AUTOMORPHISMS OF GENERALIZED THOMPSON GROUPS

MATTHEW G. BRIN AND FERNANDO GUZMÁN

Department of Mathematical Sciences State University of New York at Binghamton Binghamton, NY 13902-6000 USA August 28, 1997

Contents

0. Introduction 1 1. The classes 5 2. The Thompson groups 10 3. PL automorphisms 20 4. Relations among the automorphism groups 25 5. Restrictions on automorphisms 30 6. On the structure of Aut(A_{n}) and Aut(B_{n}) 40 7. Questions 47 References 49

0. Introduction

0.1. Results. We study the automorphisms of some generalizations of Thompson’s groups and their underlying structures. The automorphism groups of two of Thompson’s original groups were analyzed in [2] and were shown to be “small” and “unexotic.” Our results differ sharply from [2] in that we show that the automorphism groups of the generalizations are “large” and have “exotic” elements. The term exotic is from [1] and is explained later in this introduction.

Richard J. Thompson introduced the triple of infinite groups F ⊆ T ⊆ G in the 1960s and showed that they have several interesting properties. These groups were later generalized to infinite groups F_{n,∞} ⊆ F_{n} ⊆ T_{n,r} ⊆ G_{n,r} for n ≥ 2 and r ≥ 1 for which F = F_{2,∞} = F_{2}, T = T_{2,1} and G = G_{2,1}. Further background is given in the second part of this introduction.

The automorphism groups Aut(F) and Aut(T) were analyzed in [2]. In this paper, we study Aut(F_{n,∞}), Aut(F_{n}) and Aut(T_{n,n−1}) for n > 2. We find that all three contain exotic elements, and that the complexity increases as n increases. We also find that the structures of Aut(F_{n}) and Aut(T_{n,n−1}) are closely related, while differing significantly from the structure of Aut(F_{n,∞}). This difference increases as n increases. Our analysis is only partial in that we discover large subgroups of these groups containing exotic elements, but we do not know whether the subgroups are proper. Automorphism groups of a class of groups that include the Thompson groups were studied in the unpublished notes [1] where the structures were analyzed modulo the unanswered question of whether exotic automorphisms could exist.

We now give more details. Throughout the paper, functions will be written to the right of their arguments and composition will proceed from left to right.

We give two fundamental facts that underly our analyses. The first is that F_{n,∞} and F_{n} act faithfully by homeomorphisms on the real line R, and T_{n,r} acts faithfully by homeomorphisms on the circle S_{r} = R/rZ of length r. (The groups G_{n,r}, not studied, act on the Cantor set.) Further, all the homeomorhpisms used in the representations satisfy certain “local structure conditions.” That is, for each n ≥ 2, the representations use only homeomorphisms that

1. preserve orientation,

2. are piecewise linear (PL) ,

3. have slopes restricted to ⟨n⟩, the multiplicative, cyclic subgroup of the positive reals generated by n,

4. have “breaks” (discontinuities of slope) restricted to a discrete subset of Z[\frac{1}{n}], the ring of fractions with denominators in ⟨n⟩, and

5. preserve Z[\frac{1}{n}] as a set. In fact, T_{n,r} is exactly the set of self homeomorphisms of S_{r} that satisfy 1–5.

For F_{n,∞}, there is another restriction. There is a surjective ring homomorphism ϕ_{n} : Z[\frac{1}{n}] → Z_{n−1} defined by (a/n^{b})ϕ_{n} ≡ a (mod n − 1) (see Section 1) that extends modding out by n − 1. If we let Δ_{n} denote the kernel of ϕ_{n}, then the homeomorphisms used in the representation of F_{n,∞} satisfy 1–4 above and

5′. preserve Δ_{n} as a set.

See Section 2 for full definitions of the generalized Thompson groups.

The second fundamental fact is that the actions above are highly transitive. Because of this, a theorem of McCleary-Rubin [20] applies which allows us to identify the automorphism group of a generalized Thompson group H with its normalizer N(H) in the full homeomorphism group of the real line or circle. Thus we can identify an automorphism with a homeomorphism that normalizes the group in question and discuss properties of the homeomorphism (e.g., satisfies properties 1–5 above) as if they were properties of the automorphism. This practice starts with the next paragraph.

It is very easy to analyze the automorphisms of T_{n,n−1} and F_{n} that satisfy 1–5 and the automorphisms of F_{n,∞} that satisfy 1–4 and 5′. The automorphisms that are PL can also be fairly completely analyzed. Automorphisms that are not PL are labeled exotic in [1].

The structure of exotic automorphisms depends only on the local structures of the Thompson groups and the transitivity properties that bring in the McCleary-Rubin Theorem. Thus it suffices to study just one group with each of these sets of properties, and we choose the largest available. We thus invent “containers” where the containing group for groups with the properties of F_{n} is A_{n}(R) consisting of all self homeomorphisms of R or S_{r} that satisfy 1–5, and the containing group for groups with the properties of F_{n,∞} is B_{n}(R) consisting of all self homeomorphisms of R that satisfy 1–4 and 5′. Note that A_{2}(R) = B_{2}(R). The group T_{n,n−1} can serve as its own containing group.

The generalized Thompson groups are countable while A_{n}(R) and B_{n}(R) are uncountable. However, the outer automorphism groups Out(A_{n}(R)) and Out(B_{n}(R)) are countable, and each of Aut(F_{n}) and Aut(F_{n,∞}) can be largely determined from Out(A_{n}(R)) and Out(B_{n}(R)) respectively. Conversely, we use our knowledge of the Thompson groups to help understand the outer automorphisms of the containing groups. It is a consequence of [2, Lemma 3.7] that Out(A_{n}(R)) and Out(T_{n,n−1}) are isomorphic, so the role of T_{n,n−1} as a containing group will be brief.

In [2] it is shown that Out(T) = Out(A_{2}(R)) is the cyclic group of order 2 (generated by a reflection). From there it is shown that the automorphism groups of F and T are “small” and easy to describe. In particular, an index 2 subgroup in each of Aut(F) and Aut(T) consists entirely of elements that satisfy 1–5 above.

For n > 2, we show that B_{n}(R) is a characteristic subgroup of index n − 1 in A_{n}(R) and that the automorphism groups (normalizers) satsify Aut(A_{n}(R)) ⊆ Aut(B_{n}(R)). Thus there is a subgroup Out_{∗}(A_{n}(R)) of Out(B_{n}(R)) that surjects to Out(A_{n}(R)) with kernel of order n − 1.

For n not a prime, there are PL automorphisms of A_{n}(R) and B_{n}(R) that are not in A_{n}(R) and B_{n}(R) in that slopes outside of ⟨n⟩ are used. We can define Aut_{P L}(A_{n}(R)) and Aut_{P L}(B_{n}(R)) as those automorphisms that are PL, with images Out_{P L}(A_{n}(R)) and Out_{P L}(B_{n}(R)) in the outer automorphism groups.

We can now indicate some of our results.

I. Aut_{P L}(A_{n}(R)) = Aut_{P L}(B_{n}(R)) and the structure is describable modulo

difficult questions in number theory (Section 3 and Theorem 6.1.4). II. Aut(B_{n}(R)) is complete (has trivial center and trivial outer automorphism

group so that Aut^{2}(B_{n}(R)) = Aut(B_{n}(R))), but we know nothing about Aut^{2}(A_{n}(R)) (to the extent of not knowing if it is contained in Aut(B_{n}(R))) (Theorems 2.4.3 and 6.1.9).

III. For n > 2, Out_{P L}(A_{n}(R)) is of infinite index in Out(A_{n}(R)) and in particular

intersects trivially an isomorphic copy of F_{2} (Theorem 6.1.5). IV. For n > 2, Out_{∗}(A_{n}(R)) is of infinite index in Out(B_{n}(R)) and in particular

intersects trivially an isomorphic copy of F_{m,∞} for each 2 ≤ m < n (Theorems 6.1.6(d) and 6.3.2).

V. For n ≥ 4, there is an element of order n−2 in Out(B_{n}(R)) (Theorem 6.1.10). VI. For all n > 2, F = F_{2} = F_{2,∞} contains isomorphic copies of F_{n} and F_{n,∞}

(Proposition 2.5.2). Item VI is completely different from the others and its relevance(?) is touched

on in the second part of the introduction. It is elementary but seems not to be known.

In IV, Out_{∗}(A_{n}(R)) actually has trivial intersection with large parts of copies of Aut(F_{m,∞}) for 2 ≤ m < n − 1.

0.2. Background and remarks. Thompson invented the groups F ⊆ T ⊆ G in connection with studies in logic. Later, Thompson established T and G as the first known examples of infinite, finitely presented, simple groups [25] (see [7] for published details). Higman [18] defined an infinite family of finitely presented, infinite groups G_{n,r} with G = G_{2,1} and showed that G_{n,r} is simple when n is even, and that it contains a simple subgroup of index 2 when n is odd.

Brown and Geoghegan [6] established F as the first known example of a torsion free group of type F P_{∞} and not of type F P. Later Brown [5] defined infinite families of groups F_{n,∞} ⊆ F_{n} ⊆ T_{n,r} contained in G_{n,r} with F = F_{2} = F_{2,∞} and T = T_{2,1} with the F_{n} and F_{n,∞} torsion free. Brown showed that all F_{n}, F_{n,∞}, T_{n,r} and G_{n,r} are of type F P_{∞} and not of type F P.

Thompson’s groups have also appeared in homotopy theory [12], dynamical systems [14], the algebra of string rewriting [8], and diagram groups over semi-group presentations [16]. In abstract measure theory, Geoghegan has conjectured (see [4] or [13, Question 13]) that F might be a counterexample to the conjecture that any finitely presented group with no non-cyclic free subgroup is amenable (admits a bounded, non-trivial, finitely additive measure on all subsets that is invariant under left multiplication). The contents and bibliographies of [2, 3, 5, 7, 8, 16, 19] give access to all that the authors know concerning Thompson’s groups and their connections to other areas of mathematics.

Our results give no information on the amenability question, but they do raise interesting parallels between Thompson’s groups and free groups. The free group on n > 1 generators has a complete automorphism group [10, 11], and the corresponding outer automorphism group is large and contains copies of the automorphism groups of the free groups on m < n generators. Also, the free group on 2 generators contains copies of all countable free groups. These observations can be compared with II, IV and VI above.

The authors are rather surprised with V. We have no idea if there is an upper bound on the order of torsion elements in the outer automorphism groups.

Item III contains the main divergence with the results of [2]. It implies that for n > 2 there are large groups of automorphisms of the Thompson groups that do not share the local structure of the elements of the Thompson groups and in fact are not even PL. This item proves the existence of exotic automorphisms of Thompson’s groups. In [1] exotic endomorphisms are found: non-PL homeomorphisms of R that conjugate the Thompson groups into themselves, but not onto.

Item IV shows that very closely related groups (one a finite index, characteristic subgroup of the other) can have very different automorphism groups (one is of infinite index in the other). This phenonemon is not new, but examples require some work to construct. See Section 6.

We have small things to say about other subgroups of A_{n}(R) and B_{n}(R). We point out that certain subgroups that capture enough of the local structure of the elements will have their automorphism groups contained in Aut(A_{n}(R)) and Aut(B_{n}(R)). Thus we discuss a class of groups associated to each of A_{n}(R) and B_{n}(R). We have nothing to say about T_{n,r} when r is not congruent to 0 modulo n − 1 since in this case, Out(A_{n}(R)) and Out(T_{n,r}) are hard to compare.

When n − 1 has non-trivial divisors, there are containing groups between A_{n}(R) and B_{n}(R). It is clear that some of our techniques could be applied to these groups, but we do not do so here.

The results in this paper were obtained by exploiting various algebraic and geometric relationships between the various Thompson groups. For example, the algebraic Proposition 4.1.2 and geometric Lemma 4.1.5 combine to yield Theorem 4.1.6. A more purely geometric point of view is taken in [2].

The unpublished notes [1] of Bieri and Strebel contain much material about the automorphisms of groups of piecewise linear homeomorphisms of the real line. There is overlap of varying amounts between the material in [1] and material below in Sections 1, 2.6 and 3 and some material in Section 6 that ties our various results together. The parts of the current paper that do not overlap with [1] are Sections 4 and 5. The contents of the various sections are discussed briefly below. See [2, Section 1.3] for a discussion of material in [1] that complements our results. There are also some details from a very small part of [1] given in an appendix to [24].

0.3. The structure of the paper. We combine algebraic and geometric analysis. The algebra is easier to work with on R and the geometry is easier to work with on the circle. The results of [2] allow us to move back and forth between the two settings. Most of the paper takes place on the line with a few short trips to the circle to establish some key geometric facts. By and large the algebra furnishes examples and the geometry shows that the examples are non-trivial.

The point of view of [2] is adopted here that the groups we work with consist of locally defined homeomorphisms and it is the local properties that determine the automorphisms. It is for this reason that we define two classes of groups with the local properties of F_{n} and F_{n,∞} and work with A_{n}(R) and B_{n}(R), the largest members of each class. The classes are introduced in Section 1 and the Thompson groups are introduced in Section 2. The piecewise linear automorphisms are analyzed in Section 3. Techniques for constructing automorphisms are developed in Section 4. The emphasis to this point is the existence of automorphisms. Section 5 develops restrictions on the automorphisms. From the results in Section 5 all “negative” results are obtained: certain automorphisms are not PL, certain automorphisms of B_{n} are not automorhisms of A_{n}, etc. Results are gathered in Section 6, and some questions are raised in Section 7.

1. The classes

1.1. The class “containers”. We define the groups that contain all the groups in their class.

For n ≥ 2 an integer, let Z[\frac{1}{n}] = \{an^{b}|a, b ∈ Z\} and let ⟨n⟩ = \{n^{k}|k ∈ Z\}. For a PL function f from R to itself, a break of f is an x in the interior of the domain of f at which f′ is discontinuous.

Definition 1.1.1. Let n ≥ 2 be an integer. Let A_{n}(R) be the set of all homeomorphisms h : R → R so that

1. h is orientation preserving,

2. h is PL,

3. the slopes of h are all in ⟨n⟩,

4. the breaks of h are in a discrete subset of of Z[\frac{1}{n}], and

5. (Z[\frac{1}{n}])h = Z[\frac{1}{n}]. Item 1 follows from item 3, but is included for emphasis. The fact that Property 5 can be weakened to (Z[\frac{1}{n}])h ⊆ Z[\frac{1}{n}] in the presence of Property 3 will be of no use in this paper.

Lemma 1.1.2. The set A_{n}(R) is a group, and so is the set of all homeomorphisms h : R → R that satisfy Property 1, 2, 4 and 5 of Definition 1.1.1.

Proof. That Properties 1–2 are preserved under inversion and composition is standard. That Property 5 is preserved by inversion and composition is immediate. That Property 4 is preserved by inversion follows from Property 5, and that Property 4 is preserved by composition follows from the chain rule and Property 5. That Property 3 is preserved by composition and inversion follows from the chain rule.

For n ≥ 2 an integer, define ϕ_{n} : Z[\frac{1}{n}] → Z_{n−1} by (an^{b})ϕ_{n} = a (mod n − 1), and for f ∈ A_{n}(R), let fρ_{n} = (xf)ϕ_{n} − (x)ϕ_{n} for some x ∈ Z[\frac{1}{n}].

Lemma 1.1.3. For n ≥ 2 an integer,

(a) ϕ_{n} is well defined and a surjective ring homomorphism,

(b) if f ∈ A_{n}(R) and x, x′ ∈ Z[\frac{1}{n}], then (xf)ϕ_{n} − xϕ_{n} = (x′f)ϕ_{n} − x′ϕ_{n},

(c) ρ_{n} : A_{n}(R) → Z_{n−1} is a well defined, surjective homomorphism.

Proof. (a) The function ϕ_{n} is well defined since an^{b} = a′n^{b′} with all of a, b, a′, b′ integers implies that a = a′ = 0 or a/a′ is an integral power of n and we know n ≡ 1 (mod n − 1). It is a ring homomorphism for the same reason. That it is surjective is trivial.

(b) Let x = x_{0} < x_{1} < x_{2} < ⋯ < x_{n} < x_{n+1} = x′ where x_{1}, …, x_{n} are the breaks of f between x and x′. The lemma follows if it is true that (x_{i}f)ϕ_{n} − x_{i}ϕ_{n} = (x_{i+1}f)ϕ_{n} − x_{i+1}ϕ_{n} so we assume that there are no breaks between x and x′. We have x′f − xf = n^{k}(x′ − x), and since ϕ_{n} is a ring homomorphism taking n to 1, we get (x′f)ϕ_{n} − (xf)ϕ_{n} = x′ϕ_{n} − xϕ_{n} which implies the result.

(c) That ρ_{n} is well defined follows from (b). It is surjective since all translations by integers are in A_{n}(R). Now

(gf)ρ_{n} = (xgf)ϕ_{n} − xϕ_{n}
= (xgf)ϕ_{n} − (xg)ϕ_{n} + (xg)ϕ_{n} − xϕ_{n}
= fρ_{n} + gρ_{n}

.

For n ≥ 2 an integer, let Δ_{n} be the kernel of ϕ_{n} : Z[\frac{1}{n}] → Z_{n−1}. For an f ∈ A_{n}(R), we think of fρ_{n} as the amount that f rotates the cosets of Δ_{n} in Z[\frac{1}{n}].

Let B_{n}(R) be the kernel of ρ_{n} : A_{n}(R) → Z_{n−1}. The following is immediate from Lemma 1.1.3(c).

Lemma 1.1.4. For n ≥ 2 an integer, |A_{n}(R) : B_{n}(R)| = n − 1.

For a real r > 0, we let S_{r} = R/rZ be the circle of length r. If r ∈ Z[\frac{1}{n}], then rZ ⊆ Z[\frac{1}{n}], and we will denote the quotient Z[\frac{1}{n}]/rZ ⊆ S_{r} by Z[\frac{1}{n}]. If r ∈ Δ_{n}, then rZ ⊆ Δ_{n} and ϕ_{n} induces a surjective homomorphism ϕ_{n} : Z[\frac{1}{n}]/rZ → Z_{n−1}. From now on we will let context determine whether we are in S_{r} or R.

For a real r > 0, r ∈ Z[\frac{1}{n}], we let A_{n}(S_{r}) be those self homeomorphisms of S_{r} that satisfy the 5 properties in Definition 1.1.1. An argument similar to the proof of Lemma 1.1.3(c) shows the following.

Lemma 1.1.5. For a real r > 0, r ∈ Δ_{n}, there is a well defined homomorphism ρ_{n} : A_{n}(S_{r}) → Z_{n−1} given by fρ_{n} = (xf)ϕ_{n} − xϕ_{n} for any x ∈ Z[\frac{1}{n}].

Note that in the proof of Lemmas 1.1.3(b) and 1.1.5, we do not need f to be a homomorphism, and that fρ_{n} is well defined for any continuous self-map of R or S_{r} that satisfies 1–5 in Definition 1.1.1.

For a real r > 0, r ∈ Δ_{n}, we let B_{n}(S_{r}) be the kernel of ρ_{n} : A_{n}(S_{r}) → Z_{n−1}.

1.2. Transitivity and the equality of automorphism and normalizer.

Lemma 1.2.1. For n ≥ 2 an integer, let x_{1} < x_{2} < ⋯ < x_{n} and y_{1} < y_{2} < ⋯ < y_{n} all be in Z[\frac{1}{n}] such that x_{i}ϕ_{n} = y_{i}ϕ_{n} for all i with 1 ≤ i ≤ n. Then there is an f ∈ B_{n}(R) so that x_{i}f = y_{i} for each i with 1 ≤ i ≤ n.

Proof. By putting together pieces of a function with infinite portions of slope 1, it is clear that the lemma follows if we show that there is a function taking [x_{1}, x_{2}] onto [y_{1}, y_{2}] satisfying all the local properties of elements of B_{n}(R). We can divide the interval [x_{1}, x_{2}] into subintervals of length n^{k} for some integer k and we can do the same to [y_{1}, y_{2}] with intervals of length n^{l} for some integer l. Since (y_{1} − x_{1})ϕ_{n} = 0 = (y_{2}−x_{2})ϕ_{n}, we have (y_{2}−y_{1})ϕ_{n} = (x_{2}−x_{1})ϕ_{n} and thus the number of intervals used in the subdivision of [x_{1}, x_{2}] is equivalent to the number of intervals in the subdivision of [y_{1}, y_{2}] modulo n − 1. We can increase the number of intervals in a subdivision by any multiple of n − 1 by taking intervals in the subdivision and subdividing them (repeatedly if necessary) into n intervals of equal length. This preserves the property that each interval has length an integral power of n. In this way we can get subdivisions of [x_{1}, x_{2}] and [y_{1}, y_{2}] that have the same number of intervals. Now a function can be defined from [x_{1}, x_{2}] onto [y_{1}, y_{2}] that takes the intervals in the subdivision of [x_{1}, x_{2}] in order to those of the subdivision of [y_{1}, y_{2}] so as to be affine on each interval. The properties of elements of A_{n}(R) are automatically satisfied, and the extra property required to be in B_{n}(R) is guaranteed by the fact that x_{1}ϕ_{n} = y_{1}ϕ_{n}.

The techniques in the proof of previous lemma yield the next lemma.

Proposition 1.2.2. Let −∞ ≤ a < b ≤ ∞ be extended integers and let {x_{i}} and \{y_{i}\}, a < i < b, be discrete sets in Z[\frac{1}{n}] which satisfy x_{i} < x_{i+1} y_{i} < y_{i+1}, and (y_{i}−x_{i})ϕ_{n} = (y_{i+1}−x_{i+1})ϕ_{n} for each i with a < i < i+1 < b. Assume that for each j for which a < 2j < 2j + 1 < b that there is a homeomorphism f_{j} from [x_{2j}, x_{2j+1}] onto [y_{2j}, y_{2j+1}] satisfying the 5 properties in Definition 1.1.1. Then there is a function f ∈ A_{n}(R) that agrees with f_{j} for each j with a < 2j < 2j + 1 < b.

Let D be a subset of R and let a group G act on R by order preserving homeomorphisms. We say that G acts o-k-transitively on D if for for every x_{1} < ⋯ < x_{k} and y_{1} < ⋯ < y_{k} in D, there is an f ∈ G so that x_{i}f = y_{i} for each i with 1 ≤ i ≤ k. For a subset D of S_{r}, we say that a group G that acts on S_{r} by orientation preserving homeomorphisms acts o-k-transitively on D if for every x_{1}, …, x_{k} and y_{1}, …, y_{k} in D with x_{1} < ⋯ < x_{k} and y_{1} < ⋯ < y_{k} in some closed interval in S_{r} with the order inherited from the projection R → S_{r} = R/rZ, there is an f ∈ G so that x_{i}f = y_{i} for each i with 1 ≤ i ≤ k.

For an f ∈ Homeo(R), we let the support of f be the set of all x ∈ R for which xf ≠ x. If G acts on R by order preserving homeomorphisms, then we say that G has a non-trivial element of bounded support if for some f ∈ G not the identity, f is fixed off some compact subset of R. If G acts on S_{r} by orientation preserving homeomorphisms, then we say that G has a non-trivial element of bounded support if for some f ∈ G not the identity, f is fixed on some open set.

Lemma 1.2.3. The groups A_{n}(R), A_{n}(S_{r}), (r ∈ Z[\frac{1}{n}]), B_{n}(R) and B_{n}(S_{r}), (r ∈ Δ_{n}), all have non-trivial elements of bounded support and act o-k-transitively on Z[\frac{1}{n}] or Δ_{n} (in the appropriate domain) for all k.

Proof. This follows from Lemma 1.2.1 and an easily proven corresponding lemma for S_{r}.

Theorem 1.2.4 (McCleary-Rubin [20]). Let G act on R or S_{r} by orientation preserving homeomorphisms. Assume that G has a non-trivial element of bounded support. If G acts on R, assume that G acts o-2-transitively on a dense subset of R. If G acts on S_{r}, assume that G acts o-3-transitively on a dense subset of S_{r}. Then for each automorphism α of G, there is a unique self homeomorphism h of R or S_{r} (whichever is appropriate) so that fα = h^{−1}fh for every f ∈ G.

1.3. The two classes and their automorphisms. Let the class [A_{n}] consist of all subgroups G of A_{n}(R) or A_{n}(S_{r}), r ∈ Z[\frac{1}{n}], that have a non-trivial element of bounded support, that act o-2-transitiviely on R or o-3-transitively on S_{r}, and that use all the germs of A_{n}(X), X = R or X = S_{r}, in that for each x ∈ X and each f ∈ A_{n}(X), there is a g ∈ G so that f = g on some open neighborhood of x.

Let the class [B_{n}] consist of all subgroups G of B_{n}(R) or B_{n}(S_{r}), r ∈ Δ_{n}, that have a non-trivial element of bounded support, that act o-2-transitively on R or o-3-transitively on S_{r}, and that use all the germs of B_{n}(X), X = R or X = S_{r}, in that for each x ∈ X and each f ∈ B_{n}(X), there is a g ∈ G so that f = g on some open neighborhood of x.

We immediately get the following.

Lemma 1.3.1. The groups A_{n}(R), A_{n}(S_{r}), (r ∈ Z[\frac{1}{n}]), are in the class [A_{n}], and the groups B_{n}(R) and B_{n}(S_{r}), (r ∈ Δ_{n}), are in the class [B_{n}].

Let Homeo(X) be the group of homeomorphisms of the topological space X. If G is a subgroup of Homeo(X), then let N(G) denote the normalizer of G in Homeo(X).

Lemma 1.3.2. Let X = R or X = S_{r}, and let C = A or C = B. Then the following hold.

(a) For G in the class [C_{n}], taking h ∈ N(G) to conjugation of G by h is an isomorphism from N(G) to Aut(G).

(b) If G is in [C_{n}], we have N(G) ⊆ N(C_{n}(X)).

(c) G ∈ [C_{n}] is characteristic in C_{n}(X) if and only if N(G) = N(C_{n}(X)).

Proof. (a) This follows immediately from the McCleary-Rubin theorem.

(b) For h ∈ N(G), f ∈ C_{n}(X), and x ∈ X, some g ∈ G agrees with f on a neighborhood of x so that h^{−1}gh and h^{−1}fh agree on a neighborhood of xh. Since h ∈ N(G) and G ⊆ C_{n}(X), the behavior of h^{−1}gh on a neighborhood of xh satisfies the requirements in Definition 1.1.1. Thus h^{−1}fh is in C_{n}(X).

(c) If G is characteristic in C_{n}(X), then restriction gives a homomorphism from N(C_{n}(X)) to N(G) which is seen to be the bijective inverse of the inclusion in (b) by the uniqueness part of the McCleary-Rubin theorem. The converse follows from (a).

For X = R or X = S_{r}, the isomorphism between Aut(A_{n}(X)) and N(A_{n}(X)) allows us to identify Out(A_{n}(X)) with N(A_{n}(X))/A_{n}(X). Similarly for B_{n}(X). We let Out_{o}(A_{n}(X)) and Out_{o}(B_{n}(X)) be the images in Out(A_{n}(X)) and Out(B_{n}(X)) of the orientation preserving elements of N(A_{n}(X)) and N(B_{n}(X)). They are index 2 subgroups since all elements of A_{n}(X) and B_{n}(X) preserve orientation and x ↦ −x is a normalizer on both R and S_{r}. For r ∈ R, let t_{r} be the translation xt_{r} = x + r.

The next lemma follows from [2, Lemma 3.7] and its proof.

Proposition 1.3.3. (a) The projection R → S_{r} = R/rZ induces an isomorphism Out(B_{n}(R)) → Out(B_{n}(S_{r})) for r ∈ Δ_{n}.

(b) Each element of Out_{o}(B_{n}(R)) has a representative that fixes t_{n−1}.

1.4. B_{n} is characteristic in A_{n}. We start with the geometric property that makes B_{n} characteristic.

Lemma 1.4.1. With X either R or S_{n−1}, B_{n}(X) is the subgroup of A_{n}(X) generated by elements with non-empty open sets of fixed points.

Proof. Elements fixed on non-empty open sets are all in the kernel of ρ_{n} giving one containment. When X = R, and an α ∈ B_{n}(X) is given, then Proposition 1.2.2 gives a β ∈ A_{n}(X) that is the identity off some finite interval that agrees with α on an open set. Now α = ((αβ^{−1})β and both β and αβ^{−1} have non-empty open sets of fixed points. When X = S_{n−1}, an open set U is chosen so small that the closure of U ∪ Uα is contained in an open interval I in S_{n−1} whose complement contains a non-empty open set. Now β can be chosen with support in I to agree with α on U.

Proposition 1.4.2. With X either R or S_{n−1}, the normalizer of A_{n}(X) is contained in the normalizer of B_{n}(X). This implies that B_{n}(X) is characteristic in A_{n}(X).

Proof. Conjugation preserves the property of being fixed on an open set. The last statement comes from the fact that the normalizers are naturally identified with the automorphism groups.

1.5. The permutation induced by a normalizer. In this section we visit the circle to obtain information that is then lifted to the real line.

Fix an integer n ≥ 2. Let Π_{n} be the permutation group on the set {0, 1, …, n − 1}. Let μ_{n} : S_{n−1} → S_{n−1} be the map defined by xμ_{n} = nx. We say that a map is a local homeomorphism if there is an open cover of the domain so that, for each U in the cover, the map restricted to U is a homeomorphism onto the image of U.

Lemma 1.5.1. The map μ_{n} is a local homeomorphism, is not a homeomorphism, and otherwise satisfies the 5 properties in Definition 1.1.1. There are n − 1 fixed points of μ_{n} and these are the n − 1 integer points in S_{n−1}. The germs of μ_{n} are contained in the germs of B_{n}(S_{n−1}). For x ∈ S_{n−1} in Z[\frac{1}{n}], the forward orbit of x under iterates of μ_{n} ends at the fixed point xϕ_{n}.

Proof. The first sentence is immediate, and the second sentence is shown by noting that the solutions to nx = x+k(n−1) in R are the integers. We already know that the germs of μ_{n} are contained in those of A_{n}(S_{n−1}). The third sentence follows because μ_{n} has fixed points, so by the remarks following Lemma 1.1.5, μ_{n}ρ_{n} = 0. For the last sentence, the action of μ_{n} on the base n expansion of x is to shift the n-ary point one position to the right and then reduce the integer part modulo (n − 1). This preserves the value of ϕ_{n}. Eventually, the characters in the fractional part are used up and an integer remains.

Lemma 1.5.2. If h is a normalizer of B_{n}(S_{n−1}), then Z[\frac{1}{n}]h = Z[\frac{1}{n}].

Proof. The germs at an x ∈ S_{n−1} of elements of B_{n}(S_{n−1}) that have a fixed point at x form a group. If x ∈ Z[\frac{1}{n}], then this group is isomorphic to Z×Z with generators being “slope n to the right of x” and “slope n to the left of x.” For any x ∉ Z[\frac{1}{n}], this group is either isomorphic to Z or is trivial. Conjugation by h induces an isomorphism from the group of germs at x of elements with fixed point x to the group of germs at xh of elements with fixed point xh. The result follows.

Lemma 1.5.3. Let h be a normalizer of B_{n}(S_{n−1}). Then the germs of both both hμ_{n}h^{−1} and h^{−1}μ_{n}h are germs of B_{n}(S_{n−1}).

Proof. This follows immediately from Lemma 1.5.1.

Proposition 1.5.4. There are homomorphims that we ambiguously refer to as π_{n} from each of N(B_{n}(S_{n−1})), N(A_{n}(S_{n−1})), Out(B_{n}(S_{n−1})), N(B_{n}(R)), N(A_{n}(R)) and Out(B_{n}(R)), to Π_{n−1}. For h in one of these groups, the equation (xϕ_{n})(hπ_{n}) = (xh)ϕ_{n} holds for any x ∈ Z[\frac{1}{n}] and gives a well defined permutation on the residues xϕ_{n}, x ∈ Z[\frac{1}{n}], in that (xh)ϕ_{n} = (yh)ϕ_{n} if and only if xϕ_{n} = yϕ_{n} for any x and y in Z[\frac{1}{n}].

Proof. Any result for N(B_{n}) will apply to N(A_{n}) since normalizers of A_{n} are normalizers of B_{n}.

We start with an h in N(B_{n}(S_{n−1})). Since the forward orbit of each x ∈ Z[\frac{1}{n}] ends in the fixed point i = xϕ_{n}, we have that the (n − 1) sets of pre-images of the fixed points under iterates of μ_{n} are exactly the (n − 1) classes of points in Z[\frac{1}{n}] with the same values of ϕ_{n}. These are carried to the (n − 1) sets of pre-images of the fixed points under iterates of h^{−1}μ_{n}h. However, since the germs of h^{−1}μ_{n}h are in the germs of B_{n}(S_{n−1}), each of these sets consists entirely of elements of Z[\frac{1}{n}] with a common value of ϕ_{n}. Since Z[\frac{1}{n}]h = Z[\frac{1}{n}], these sets cover all of Z[\frac{1}{n}] and therefore all of the (n − 1) values of ϕ_{n} on Z[\frac{1}{n}]. Thus the (n − 1) sets of preimages of the fixed points under iterates of h^{−1}μ_{n}h are exactly the (n − 1) classes of points in Z[\frac{1}{n}] with the same values of ϕ_{n}. The fact that π_{n} is a homomorphism follows from the fact that i(hπ_{n}) = (ih)ϕ_{n}. We get an induced homomorphism on Out(B_{n}(S_{n−1})) since B_{n}(S_{n−1}) is contained in the kernel of π_{n}.

To get π_{n} defined on N(B_{n}(R)), we note that Proposition 1.3.3(b) implies that any h in N(B_{n}(R)) induces a permutation on the values of ϕ_{n} modulo an element of B_{n}(R)). However, elements of B_{n}(R)) act trivially on the values of of ϕ_{n}. Thus we get a well defined hπ_{n}. That this is a homomorphism and that we get an induced homomorphism on Out(B_{n}(R)) follows as on S_{n−1}.

2. The Thompson groups

2.1. Definitions and presentations. We define the groups referred to as generalized Thompson groups in the introduction, and we prove basic facts about them. The material here (through Proposition 2.4.2) can be found in [5, pp. 51–63] where the Thompson groups are defined as subgroups of automorphism groups of certain algebras and it is proven that they are isomorphic to the groups defined below. Since the basic setting, the arrangement of material and some of the proofs are different from what is found in [5], and since we will need some of the details of arguments, we give full proofs. See the references in [2] for other papers that treat all or parts of this material when n = 2. Our notation is largely, but not entirely consistent with that of [5].

Fix an integer n ≥ 2. For a real i, let g_{i} be the self homeomorphism of R defined by

xg_{i} = \begin{cases}x & x < i, \\ n(x − i) + i & i ≤ x ≤ i + 1, \\ x + n − 1 & x > i + 1. \end{cases}

Typically, i will be an integer. Recall that xt_{r} = x + r. Note that all the g_{i} for i ∈ Z[\frac{1}{n}] are in B_{n}(R) and that t_{r} is in A_{n}(R) if r ∈ Z[\frac{1}{n}] and in B_{n}(R) if r ∈ Δ_{n}.

The Thompson groups will be defined by fixing an n and chosing various of the homeomorphisms g_{i} and t_{r} as generators. Parts of this paper make use of the relationships that exist between the Thompson groups for different values of n. When such discussions take place, it will be necessary to distinguish between the g_{i} for various values of n. In those places, we will use g_{n,i} to refer to the homeomorphism g_{i} that is defined in the previous paragraph.

Definition 2.1.1. Fix an integer n ≥ 2. If Y is a set of elements in a group G, then we use ⟨Y ⟩ to denote the subgroup of G generated by the elements in Y. Using the elements of A_{n}(R) and B_{n}(R) described above we establish notation for certain subgroups of A_{n}(R) and B_{n}(R).

\begin{aligned}F_{n,∞} & = ⟨t_{n−1}, …, g_{−1}, g_{0}, g_{1}, … ⟩ \\ F_{n} & = ⟨t_{1}, …, g_{−1}, g_{0}, g_{1}, … ⟩ \\ F_{n,i} & = ⟨g_{i}, g_{i+1}, … ⟩ \\ F_{n,−∞} & = ⟨…, g_{−1}, g_{0}, g_{1}, … ⟩ \end{aligned}

The group F_{n,i} is not the same as the group F_{n,r} in [5]. Our group F_{n,∞} corresponds more closely to the embedding of F_{n,∞} in F_{n} in [5, Prop. 4.3] than the definition of F_{n,∞} in [5, Prop. 4.1].

The next item is established by direct checking. Shortly, we will see that the relations given there are sufficient to give presentations of the groups defined above.

Lemma 2.1.2. Each of the following relations

g_{i}^{−1}g_{j}g_{i} = g_{j+n−1}  for all i < j
t_{n−1}^{−1}g_{j}t_{n−1} = g_{j+n−1}  for all j
t_{1}^{−1}g_{j}t_{1} = g_{j+1}  for all j

is satisfied by the generators of F_{n,∞}, F_{n}, F_{n,i} and F_{n,−∞} whenever the generators in the relation are in the group in question.

Thus the groups F_{n,∞}, F_{n} and F_{n,i} are finitely generated and:

\begin{aligned}F_{n,∞} & = ⟨t_{n−1}, g_{0}, g_{1}, …, g_{n−2}⟩ \\ F_{n} & = ⟨t_{1}, g_{0}⟩ \\ F_{n,i} & = ⟨g_{i}, g_{i+1}, …, g_{i+n−1}⟩. \end{aligned}

The group F_{n,−∞} is not finitely generated since it is the union of

F_{n,0} ⊊ F_{n,−1} ⊊ F_{n,−2} ⊊ · · · .

Definition 2.1.3. Let F_{n,∞}^{∗}, F_{n}^{∗}, F_{n,i}^{∗} and F_{n,−∞}^{∗} be the groups presented with generators as given in Definition 2.1.1 of the corresponding unstarred group and all relations in Lemma 2.1.2 that apply to the generators of the group.

Lemma 2.1.4 (Semi-normal form). An element of F_{n,∞}^{∗}, F_{n}^{∗}, F_{n,i}^{∗} or F_{n,−∞}^{∗} can be written as P N^{−1} where both P and N have the form

t_{r}^{ϵ}g_{i_{1}}^{δ_{1}} ⋯ g_{i_{k}}^{δ_{k}}

where ϵ ≥ 0 and is positive only if appropriate to the group, r is n − 1 or 1 as appropriate to the group, each δ_{j} ≥ 1, and i_{1} < i_{2} < ⋯ < i_{k}.

Proof. The relations of Lemma 2.1.2 imply the following:

g_{j}^{ϵ}g_{i} = g_{i}g_{j+n−1}^{ϵ}  for all i < j
g_{i}^{−1}g_{j}^{ϵ} = g_{j+n−1}^{ϵ}g_{i}^{−1}  for all i < j
g_{j}^{ϵ}t_{n−1} = t_{n−1}g_{j+n−1}^{ϵ}  for all j
t_{n−1}^{−1}g_{j}^{ϵ} = g_{j+n−1}^{ϵ}t_{n−1}^{−1}  for all j
g_{j}^{ϵ}t_{1} = t_{1}g_{j+1}^{ϵ}  for all j
t_{1}^{−1}g_{j}^{ϵ} = g_{j+1}^{ϵ}t_{1}^{−1}  for all j

where ϵ is +1 or −1 in each relation. In words, the first equation above says that positive powers of g_{i} can move to the left of a g_{j}^{ϵ} with j > i at the expense of raising j. There are similar wordings of the remaining equations. The achievement of the semi-normal form now follows by sliding generators to their appropriate positions.

The above is not a normal form. For example, when n = 2, we have g_{0}^{2}g_{2}g_{0}^{−2} = g_{0}g_{1}g_{0}^{−1}. We can achieve a normal form by making suitable restrictions, but we have no need for this. A discussion for n = 2 is in [6] and [7].

Proposition 2.1.5. In F_{n,∞}^{∗}, F_{n,i}^{∗} and F_{n,−∞}^{∗} any non-trivial normal subgroup contains the commutator subgroup. In F_{n}^{∗}, any non-trivial normal subgroup contains the commutators of all pairs in {t_{n−1}, …, g_{−1}, g_{0}, g_{1}, … }.

Proof. Let a non-trivial element in a normal subgroup K have semi-normal form P N^{−1}. By conjugating (if necessary), we can assume that P and N start with different generators. By inverting (if necessary), we can assume that P starts with t_{r} (with an appropropriate r) or with a g_{i} of lower subscript than the generator at the start of N. Let the generator at the start of P be γ and let P = γ^{k}P_{1} where P_{1} does not start with a non-zero power of γ.

From this point to the end of the proof, all statements of equality, triviality or commutativity are modulo the normal subgroup K. Now either γ^{k} is trivial, or γ^{k} equals a word w = P_{1}N^{−1}, in generators of “higher subscript” than that of γ (all g_{i} have “higher subscript” than that of t_{r}). We claim: (∗) These facts imply that γ^{k} commutes with all g_{i} and with t_{n−1} if applicable.

We can assume that γ^{k} is not trivial. For an integer q, let w_{q} be w with the subscripts of all its generators raised uniformly by q(n−1). Conjugating γ^{k}w^{−1} ∈ K by powers of γ, gives γ^{k} = w_{q} for all q ≥ 0. Any g_{i} or t_{n−1} conjugates some w_{q} to w_{q+1} for some sufficiently large q, and so commutes with γ^{k}. This proves (∗). (∗∗) If g_{j} has subscript higher than that of γ, then γ^{k} conjugates g_{j} to some g_{m} with m > j and we have g_{j} = g_{m} by the conclusion of (∗). This says that g_{j} satisfies the hypotheses about γ^{k} in (∗), so g_{j} commutes with all g_{i} and with t_{n−1} if applicable.

If g_{j} is arbitrary and α = g_{i} for some i < j or α = t_{n−1} if applicable, then α^{q} conjugates g_{j} to (g_{j})_{q} (notation as above), and α^{q+1} conjugates g_{j} to (g_{j})_{q+1}. For q sufficiently large, (g_{j})_{q} = (g_{j})_{q+1} by (∗∗). Thus α = α^{q+1}(α^{q})^{−1} commutes with g_{j} and the proof is complete.

Corollary 2.1.5.1. Each of the groups F_{n,∞}, F_{n}, F_{n,i} and F_{n,−∞} is presented by the “starred” presentation given in Definition 2.1.3.

Proof. The homomorphism taking each formal generator to the homeomorphism it denotes yields an injection since no two of the homeomorphisms g_{i} and g_{j} with i ≠ j commute.

Note that conjugating by either t_{1} or t_{n−1} induces an automorphism on each of F_{n,∞}, F_{n} and F_{n,−∞} and an endomorphism on each F_{n,i}. Conjugating by t_{1} fixes t_{1} and t_{n−1} and takes each g_{i} to g_{i+1} and conjugating by t_{n−1} fixes t_{1} and t_{n−1} and takes each g_{i} to g_{i+n−1}. We use σ_{1} to denote the automorphisms and endomorphisms induced by conjugation by t_{1} and use σ_{n−1} to denote the automorphisms and endomorphisms induced by conjugation by t_{n−1}.

The next lemma follows easily from the lemmas thus far established.

Lemma 2.1.6. The groups F_{n,∞} and all F_{n,i}, i ∈ Z, are isomorphic. The group F_{n,∞} is a semidirect product of F_{n,−∞} and Z where the action of one generator of Z on F_{n,−∞} is that of σ_{n−1}. The group F_{n} is a semidirect product of F_{n,−∞} and Z where the action of one generator of Z on F_{n,−∞} is that of σ_{1}. The groups F_{n,∞} and F_{n} are also ascending HNN extensions of each F_{n,i} where the defining endomorphisms are σ_{n−1} and σ_{1} respectivley.

2.2. Characterizing the elements of Thompson’s groups. For each of the groups defined above, we characterize the self homeomorphisms of R that are its elements. To do this we will build new machinery for describing self homeomorphisms of R. For the rest of this section, we fix n at some integer greater than 1.

A subdivision of R into closed intervals is a locally finite collection of bounded, closed intervals whose union is all of R and whose interiors are non-empty and pairwise disjoint. Note that a subdivision of R can be described completely by its set of endpoints, and that the sets of endpoints of subdivisions of R are precisely those subsets of R that are closed, discrete and that have neither an upper nor a lower bound.

Definition 2.2.1. Let the standard subdivision S of R refer to the collection of closed unit intervals [i, i+1], i ∈ Z. The intervals in the standard subdivision S will be called the standard intervals. Given a subdivision of R into closed intervals, an allowable modification of the subdivision is to replace one of the closed intervals [a, b] in the subdivision by its subdivision into the n intervals [a+(b−a)\frac{i}{n}, a+(b−a)\frac{i+1}{n}], 0 ≤ i < n. An allowable subdivision of R is one obtained from the standard subdivision S of R by a finite number of allowable modifications. Note that all but finitely many of the intervals in an allowable subdivision of R will be standard intervals. An allowable interval is an interval in some allowable subidivision of R. It is immediate that the allowable intervals are those intervals of the form [in^{k}, (i + 1)n^{k}] for some integer i and non-positive integer k. We say that an allowable subdivision of R is supported in a subset A of R if A contains all the non-standard intervals of the subdivision.

Let D and R be two allowable subdivisions of R. An isomorphism from D to R is an orientation preserving homeomorphism from R to itself that is the identity near −∞ and that takes each interval in D affinely onto an interval of R. There is only one isomorphism from a given D to given R. The definition of isomorphism could be made more flexible, the next lemma could cover more groups than F_{n,0}, and Definition 2.2.3 extended to cover these changes, but we will have no need of this.

Lemma 2.2.2. The group F_{n,0} consists of all isomorphisms between pairs of allowable subdivisions of R that are supported in [0, ∞).

We will give a constructive proof since we will have later need of the construction. We first give a lemma that will go into the proof and whose statement we will need later.

Definition 2.2.3. We call a word P in the generators g_{i}, i ≥ 0, of F_{n,0} as given in Definition 2.1.1 a positive word if all generators appear with positive exponent. Note that each of the parts P and N of a semi-normal form P N^{−1} of a word in the generators of F_{n,0} is a positive word. We will define the subdvision of R that we associate with such a positive word P. Let P = g_{i_{1}}g_{i_{2}} ⋯ g_{i_{k}}. Let the prefixes of P be P_{0} = 1, P_{1} = g_{i_{1}}, …, P_{k−1}, P_{k} = P where P_{j} = P_{j−1}g_{i_{j}}. We inductively associate subdivisions D_{0}, …, D_{k} to the prefixes where D_{0} = S. To do so we need a numbering scheme. Note that any allowable subdivision of R has 0 as an endpoint of an interval. We number the intervals of an allowable subdivision of R that is supported in [0, ∞) so that interval 0 has 0 as its left endpoint and the intervals are numbered by the integers consecutively from left to right in R. We now define D_{j} as that subdivision obtained from D_{j−1} by subdividing interval numbered i_{j} in D_{j−1} into n subintervals of equal size (here i_{j} is the subscript of the ending generator in P_{j}). The subdivision associated with each P_{j} is D_{j}. In F_{n,0}, the subscripts of all the g_{i} are non-negative, so all of the subdivisions D_{j} will be supported in [0, ∞).

Lemma 2.2.4. (a) Every allowable subdivision of R that is supported in [0, ∞) is associated with some positive word in semi-normal form in the generators of F_{n,0}.

(b) If P is a positive word in the generators of F_{n,0}, then P represents an isomorphism from the subdivision D associated with P to the standard subdivision S of R.

(c) If P N^{−1} is a semi-normal form for an element of F_{n,0}, then it represents an isomorphism from the allowable subdivision D of R supported on [0, ∞) and associated with P to the allowable subdivision R of R supported on [0, ∞) and associated with N.

Proof. We get (a) from the definitions by always choosing the leftmost interval to subdivide when there is an ambiguity as to which interval to subdivide next. We get (b) by induction on the length of P and the observation that (b) is true when P is a single generator g_{i}, and we get (c) from (b).

Proof of Lemma 2.2.2. Both containments of the claimed equality follow from the previous lemma.

Proposition 2.2.5. The group F_{n,0} consists of all elements f of A_{n}(R) that are the identity to the left of 0 and are translations by integegral multiples of n − 1 near ∞ in that there are integers i and M so that xf = x + i(n − 1) for all x > M.

Proof. The generators of F_{n,0} all fit the description in the statement and the description is preserved under inverses and composition. Thus it remains to show that all elements as described are in F_{n,0}.

Let f be as described in the statement. By the previous lemma, we must show that f is an isomorphism between two allowable subdivisions P and Q of R that are supported in [0, ∞). Near ±∞, the standard subdivision supports f as an isomorphism. We must arrange things on a compact set. There is an allowable subdivision P of S so that f is linear on each interval in P. This is done by subdividing intervals until the breaks of f are contained in the endpoints of the subdivison. Since f is the identity to the left of 0, only intervals in [0, ∞) will have to be subdivided. The image subdivision P f has its endpoints in Z[\frac{1}{n}], and it can be subdivided to a subdivision Q′ by allowable modifications so that Q′ is equal to S except on some interval [a, b], 0 ≤ a < b, a, b ∈ Z, where Q′ consists of intervals of one constant length n^{−k}. Let P′ be the subdivision Q′f^{−1} of P. Since f is linear with slope an integral power of n on each interval of P, this subdivision of P uniformly subdivides each interval of P into subintervals of length n^{−j} for an integer j that depends on the interval. It follows that P′ is an allowable subdivision of P and f is an isomorphism carried by the allowable pair (P′, Q′).

In the proof above, we did not use the fact that the translation near ∞ was by an integral multiple of n − 1 because we did not have to. Any element of A_{n}(R) that fixes 0 and is a translation by an integer near ∞ is a translation by an integral multiple of n − 1 near ∞.

The next proposition follows easily from the proposition just proven.

Proposition 2.2.6. The group F_{n,−∞} consists of all elements f of A_{n}(R) that are the identity near −∞ and that are translations by integral multiples of n − 1 near ∞. The group F_{n,∞} consists of all elements f of A_{n}(R) that are translations by integral multiples of n − 1 near ±∞ in that there are integers i, j and M so that xf = x + i(n − 1) for all x < −|M| and xf = x + j(n − 1) for all x > |M|. The group F_{n,i} consists of all elements f of F_{n,∞} that are the identity on (−∞, i). The group F_{n} consists of all elements of A_{n}(R) that are translations by integers near ±∞ in that there are integers i, j and M so that xf = x + i for all x < −|M| and xf = x + j for all x > |M|.

Since F_{n} ⊆ A_{n}(R), each f ∈ F_{n} rotates the cosets of Δ_{n} by ρ_{n}(f). Thus the values of i and j in the description of elements of F_{n} in the previous statement must be congruent modulo n − 1. This observation and the preceeding proposition yield:

Corollary 2.2.6.1. |F_{n} : F_{n,∞}| = n − 1.

Lemma 2.2.7. The group F_{n} is in [A_{n}], and the groups F_{n,∞}, F_{n,−∞} and F_{n,i} are in [B_{n}].

Proof. If C is any compact subset of R and f ∈ A_{n}(R), then Proposition 1.2.2 implies that there is a g in A_{n}(R) that agrees with f on C and is a translation by fϕ_{n} near ±∞. This suffices to put g in F_{n}, and suffices to prove that F_{n} is in [A_{n}]. Similar arguments give the rest of the statement.

2.3. Thompson’s group on a closed interval. Let F_{n,[0,n−1]} be the set of elements of A_{n}(R) that have support in [0, n − 1]. By Proposition 2.2.6, this is a subgroup of F_{n,0} and thus of B_{n}(R).

Lemma 2.3.1. There is a homeomorphism from [0, ∞) to [0, n − 1) that induces, by conjugation, an isomorphism from F_{n,0} to F_{n,[0,n−1]}.

Proof. Let D_{0} be the subdivision of [0, n−1] that consists of the n−1 standard unit intervals [i, i+1], 0 ≤ i < n−1, in [0, n−1]. Let D_{i} be obtained from the subdivision D_{i−1} by subdividing the rightmost interval of D_{i−1} into n equal intervals. Let S′ be the limit of this process. This is a subdivision of the interval [0, n − 1) into countably many allowable intervals. If we define allowable subdivisions of S′ as the result of finite sequences of allowable modifications of S′, then we can define isomorphisms between pairs of allowable subdivisions of S′.

We note that an allowable modification raises the number of intervals in given closed interval by n − 1. Thus an isomorphism between allowable subdivisions of S′ results in a shift of the intervals of S′ near n − 1 by some integral multiple k of n − 1. It is easy to show that shuch a shift is an affine function in a neighborhood of n − 1 with fixed point n − 1 and slope n^{−k}. Thus every such isomorphism represents a function in F_{n,[0,n−1]}. An argument identical to that of Proposition 2.2.5 shows that every element of F_{n,[0,n−1]} is such an isomorphism. The claimed conjugating homeomorphism in the statement of the lemma is the isomorphism taking the intervals in the standard subdivision S that are in [0, ∞) to the intervals in S′.

Corollary 2.3.1.1. The subgroup B_{n}^{0}(S_{n−1}) of B_{n}(S_{n−1}) that consists of all elements that fix 0 is isomorphic to F_{n,[0,n−1]}, to F_{n,0} and also to F_{n,∞}.

Proof. The subgroup is clearly isomorphic to F_{n,[0,n−1]}, and we use Lemmas 2.3.1 and 2.1.6.

2.4. A simple subgroup. We define two more subgroups by properties of their elements. The subgroups are not conveniently given as presentations.

Let F_{n}^{0} consist of those elements of F_{n} that are the identity near ±∞.

Lemma 2.4.1. We have F_{n}^{0} ⊆ F_{n,−∞} ⊆ F_{n,∞}. The group F_{n}^{0} consists of all f ∈ F_{n,−∞} whose every expression as a word w in the generators g_{j} has the exponent sum of w over all g_{i} equal to zero.

Proof. The containments are clear. The second claim follows because each generator g_{j} behaves as translation by n − 1 near +∞.

Let F_{n}^{s} consist of those elements f of F_{n,−∞} so that an expression of f as a word w in the generators g_{j} satisfies that for each i with 0 ≤ i < n − 1, the exponent sum of w over all generators g_{j} with j ≡ i (mod n − 1) is zero. Note that this is well defined since the relations in F_{n,−∞} preserve such sums.

Proposition 2.4.2. The following are true:

(a) F_{n}^{s} ⊆ F_{n}^{0} ⊆ F_{n,−∞} ⊆ F_{n,∞}.

(b) If g ∈ B_{n}(R) and a compact C ⊆ R are given, then there is an f ∈ F_{n}^{s} that agrees with g on C.

(c) F_{n}^{s} and F_{n}^{0} are in [B_{n}].

(d) F_{n}^{0} is characteristic in F_{n,−∞} and B_{n}(R).

(e) F_{n}^{s} is the commutator subgroup of F_{n,−∞} and F_{n}^{0}.

(f) F_{n}^{s} is characteristic in F_{n}^{0}, F_{n,−∞} and B_{n}(R).

(g) F_{n}^{s} is simple.

Proof. (a) This is immediate.

(b) As mentioned in the proof of Lemma 2.2.7, it is easy to find an f ∈ F_{n,−∞} that equals g on C. To get f into F_{n}^{s} one composes f on the right by powers of various generators g_{j}. Since one can choose the subscripts j up to integral multiples of n − 1, the subscripts can be chosen so large that each new g_{j} is the identity on C.

(c) This follows from (a) and (b).

(d) From the McCleary-Rubin Theorem, every automorphism of B_{n}(R) and every automorphism of F_{n,−∞} is realized as conjugation by a self homeomorpism of R. Such a conjugation preserves the property of being fixed off some compact set.

(e) The relations of F_{n,−∞} establish that g_{i} is conjugate to g_{j} if and only if i ≡ j (mod n−1). Modulo the commutator subgroup, there are no other relations. Thus the abelianization of F_{n,−∞} is the sum of n − 1 copies of Z and F_{n}^{s} is the kernel of the abelianization homomorphism. Thus F_{n}^{s} is the commutator subgroup of F_{n,−∞}.

Since F_{n}^{0} is characteristic in F_{n,−∞}, the commutator subgroup of F_{n}^{0} is normal in F_{n,−∞} and it lies in F_{n}^{s} the commutator subgroup of F_{n,−∞}. By Proposition 2.1.5, the commutator subgroup of F_{n}^{0} is all of F_{n}^{s} if the commutator subgroup of F_{n}^{0} is non-trivial. However this follows from (b) since F_{n}^{s} ⊆ F_{n}^{0}.

(f) This follows from (e) and (d).

(g) We can use a simplification of an argument of Higman [17]. We must show that every non-trivial normal subgroup of F_{n}^{s} contains all of F_{n}^{s} the commutator subgroup of F_{n}^{0}. That is, we must show that the commutator of any two elements of F_{n}^{0} lies in the normal closure in F_{n}^{s} of any non-trivial element of F_{n}^{s}.

Let f be non-trivial in F_{n}^{s} and let g and h be in F_{n}^{0}. There is an open set U ⊆ R so that Uf ∩U = ∅. Since g and h are in F_{n}^{0}, the union of their supports lies in some compact set C ⊆ R. Because F_{n}^{s} is in [B_{n}], it has enough transitivity properties so that there is a j ∈ F_{n}^{s} so that C ⊆ Uj. Now j^{−1}fj carries Uj to Ufj which is disjoint from Uj. Thus j^{−1}fj carries C ⊆ Uj to a set disjoint from C. The support of h conjugated by j^{−1}fj lies in Cj^{−1}fj and is disjoint from the support of g. Thus h conjugated by j^{−1}fj commutes with g, and modulo the normal closure in F_{n}^{s} of f, we have that g and h commute.

Theorem 2.4.3. The automorphism group of Aut(B_{n}(R)) equals Aut(B_{n}(R)).

Proof. By Lemmas 1.3.2(c) and 2.4.2(f), Aut(B_{n}(R)) = Aut(F_{n}^{s}), and since F_{n}^{s} is simple, [23, Theorem 13.5.10] or [22, Stmt. 6.2.1.5] give that Aut(F_{n}^{s}) is its own automorphism group.

2.5. Monomorphisms from universal properties. This short section recalls a universal property of the Thompson groups and uses it to show the existence of monomorphisms between the Thompson groups defined for different values of n, including monomorphisms from any Thompson group acting on R to F_{2,0}. In Section 4, we will derive different monomorphims between Thompson groups that allow for extensions of automorphisms. The monomorphisms in this section do not seem to have such extension properties.

The endomorphism σ_{1} : F_{n,0} → F_{n,0} taking each g_{i} to g_{i+1} has the property that σ_{1}^{n} ≠ σ_{1} but σ_{1}^{n} = σ_{1}T_{0} where T_{0} : F_{n,0} → F_{n,0} is the inner automorphism gotten by conjugating by the generator g_{n,0}. The next lemma shows that F_{n,0} is universal in this behavior. This is a trivial extension of the n = 2 case [12, p. 95] and [6, Prop. 1.1].

Lemma 2.5.1. Let the group G admit an endomorphism σ : G → G with the property that σ^{n} ≠ σ but σ^{n} = σT_{a} for some non-identity a ∈ G where T_{a} : G → G is the inner automorphism gotten by conjugating by a. Then there is a homomorphism κ : F_{n,0} → G with g_{n,i}κ = aσ^{i}, for each i ≥ 0. Further, if a fails to commute with aσ, then κ is a monomorphism.

Proof. Note that for 0 ≤ j < k we have

(aσ^{j})^{−1} (aσ^{k}) (aσ^{j}) = (aσ^{j})^{−1} (aσ^{k−j}σ^{j}) (aσ^{j})
= [a^{−1}(aσ^{k−j})a] σ^{j}
= [a^{−1} ((aσ^{k−j−1})σ) a] σ^{j}
= [(aσ^{k−j−1})σ^{n}] σ^{j}
= aσ^{k+n−1}.

Thus the images of a under non-negative powers of σ satisfy the relations of F_{n,0}. If a does not commute with aσ, then κ is a monomorphism by Proposition 2.1.5.

Proposition 2.5.2. Let 2 ≤ m < n satisfy (m−1)|(n−1) with (n−1) = (m−1)d. Then there is a monomorphism λ_{n,m} : F_{n,0} → F_{m,0} with g_{n,i}λ_{m,n} = g_{m,i}^{d} for each i ≥ 0 and there is a monomorphism λ′_{m,n} : F_{m,0} → F_{n,0} with g_{m,i}λ′_{m,n} = g_{n,di} for each i ≥ 0.

Proof. We consider λ_{n,m}. If T_{m,0} denotes conjugation by g_{m,0}, then we have σ_{1}T_{m,0} = σ_{1}^{m}. The hypotheses on m and n now imply that σ_{1}T_{m,0}^{d} = σ_{1}^{n}. But T_{m,0}^{d} is just conjugation by g_{m,0}^{d} and the images of g_{m,0}^{d} under powers of σ_{1} are just the elements g_{m,i}^{d}. Since g_{m,0}^{d} and g_{m,1}^{d} do not commute, the properties of λ_{n,m} now follow from the previous lemma.

For λ′_{m,n}, let Σ = σ_{1}^{d}. Now

ΣT_{n,0} = σ_{1}^{d}T_{n,0} = σ_{1}^{d+n−1} = σ_{1}^{d+d(m−1)} = σ_{1}^{dm} = Σ^{m}.

The properties of λ′_{m,n} follow from the previous lemma and the fact that g_{n,0} and g_{n,d} = g_{n,0}Σ do not commute.

Note that the hypothesis in the previous proposition is satisfied for all n ≥ 2 if m = 2. Further, if we send t_{1} to g_{2,0} and, for each i ≥ 0, g_{n,i} to g_{2,i+1}^{n−1}, we get a monomorphism from F_{n} into F_{2,0}. This implies that there are monomorphisms into F_{2,0} from each of the groups in Definition 2.1.1 and Section 2.3.

2.6. Applying the Thompson groups to automorphisms. We are interested in Out_{o}(A_{n}(R)) and Out_{o}(B_{n}(R)) as defined just before Proposition 1.3.3. We will show that these groups can be obtained from the Thompson groups and their automorphisms and that to a large extent, the automorphism groups of the Thompson groups can be obtained from Out_{o}(A_{n}(R)) and Out_{o}(B_{n}(R)).

Because of Proposition 1.3.3, it is convenient to look at automorphisms that commute with certain translations. We will use the notation of the next paragraph extensively. When we do, g will always be a translation.

Let G be a group of self homeomorphisms of R to which the McCleary-Rubin theorem applies and identify Aut(G) with the normalizer of G in the full group of self homeomorphisms of R. If g is a self homeomorphism of R, with g not necessarily in G, then let Aut(G, g) be the set of automorphisms of G that fix g, and let C(G, g) be the set of inner automorphisms of G that fix g (and so is the centralizer of g in G). We refer to Aut(G, g) as the automorphism group of the pair (G, g). We use Aut^{0}(G, g) and C^{0}(G, g) to denote the subgroups of the respective groups that consist of the elements that fix 0.

The automorphisms of B_{n}(R) are somewhat better behaved than those of A_{n}(R), and so we start with B_{n}(R). The following is a definition:

1 → B_{n}(R) → Aut_{o}(B_{n}(R)) → Out_{o}(B_{n}(R)) → 1.

From Proposition 1.3.3, we know that every element of Out_{o}(B_{n}(R)) has a representative the commutes with t_{n−1}. Automorphisms (normalizers) that commute with t_{n−1} preserve the defining property of F_{n,∞} within B_{n}(R). Conversely, Lemma 2.2.7 says that automorphisms of F_{n,∞} are automorphisms of B_{n}(R). Thus we obtain

1 → Aut_{B}(F_{n,∞}) → Aut_{o}(F_{n,∞}) → Out_{o}(B_{n}(R)) → 1

where Aut_{B}(F_{n,∞}) = B_{n}(R)∩Aut_{o}(F_{n,∞}). If we push further the observation that elements of Out_{o}(B_{n}(R)) have representatives the commute with t_{n−1}, we get

1 → C(B_{n}(R), t_{n−1}) → Aut(F_{n,∞}, t_{n−1}) → Out_{o}(B_{n}(R)) → 1.

We now pass to a normal subgroup of finite index. We define Aut_{π}(F_{n,∞}, t_{n−1}) and Out_{π}(B_{n}(R)) to be the kernels of the homomorphism π_{n} applied to Aut(F_{n,∞}, t_{n−1}) and Out_{o}(B_{n}(R)) repsectively. We will see in Theorem 4.4.3 that (Out_{o}(B_{n}(R)))π_{n} is all of Π_{n−1} which will show that the index of these subgroups is (n − 1)!. We have

1 → C(B_{n}(R), t_{n−1}) → Aut_{π}(F_{n,∞}, t_{n−1}) → Out_{π}(B_{n}(R)) → 1.

We can make a further reduction. Given h ∈ Aut_{π}(F_{n,∞}, t_{n−1}), we have bϕ_{n} = (0h)ϕ_{n} = 0(hπ_{n}) = 0. Thus t_{b} is in B_{n}(R), and h′ = ht_{b}^{−1} fixes 0 and represents the same element in Out_{o}(B_{n}(R)) as h. We now have

1 → C^{0}(B_{n}(R), t_{n−1}) → Aut_{π}^{0}(F_{n,∞}, t_{n−1}) → Out_{π}(B_{n}(R)) → 1.

The order in the above sequence of short exact sequences is largely forced. We cannot restrict to normalizers that fix 0 before we restrict to normalizers that have π_{n} trivial. This is because without the triviality of π_{n}, the translation t_{b} used above might not be in B_{n}(R).

The automorphisms of A_{n}(R) are better behaved in that they contain more translations, and the restriction to normalizers that fix 0 is easier to achieve. However, Proposition 1.3.3 does not apply to Out_{o}(A_{n}(R)). We will get around this by trivializing permutations once again, but here we are not dealing with a normal subgroup of Out_{o}(A_{n}(R)) since π_{n} is not well defined on Out_{o}(A_{n}(R)).

We start with the defining sequence

1 → A_{n}(R) → Aut_{o}(A_{n}(R)) → Out_{o}(A_{n}(R)) → 1.

Given h ∈ Aut_{o}(A_{n}(R)), we have b = 0h ∈ Z[\frac{1}{n}]. Thus t_{b} is in A_{n}(R), and h′ = ht_{b}^{−1} fixes 0 and represents the same element in Out_{o}(A_{n}(R)) as h. We have

1 → A_{n}^{0}(R) → Aut_{o}^{0}(A_{n}(R)) → Out_{o}(A_{n}(R)) → 1.

However, elements of A_{n}(R) that fix 0 are in B_{n}(R), so we get

1 → B_{n}^{0}(R) → Aut_{o}^{0}(A_{n}(R)) → Out_{o}(A_{n}(R)) → 1.

With the kernel in the above sequence containing only elements with trivial π_{n}, we can pass to a subgroup. We let Aut_{π}^{0}(A_{n}(R)) be the kernel of π_{n} restricted to Aut^{0}(A_{n}(R)). We let Out_{π}(A_{n}(R)) be the image of Aut_{π}^{0}(A_{n}(R)) in Out_{o}(A_{n}(R)). We will see in Theorem 6.1.8 that the indexes are not (n−1)! and are not completely known. We get

1 → B_{n}^{0}(R) → Aut_{π}^{0}(A_{n}(R)) → Out_{π}(A_{n}(R)) → 1.

We now observe that a minor change to the proof of [2, Lemma 3.7] gives the following. In the change, we can assume that the normalizer h fixes 0 and our extra assumption says that (1h)ϕ_{n} = 1.

Proposition 2.6.1. Each element of Out_{π}(A_{n}(R)) has a representative that fixes t_{1}.

Since automorphisms (normalizers) that commute with t_{1} preserve the defining property of F_{n} within A_{n}(R) and automorphisms of F_{n} are automorphisms of A_{n}(R), we can now bring in a Thompson group:

1 → Aut_{B}^{0}(F_{n}) → Aut_{π}^{0}(F_{n}) → Out_{π}(A_{n}(R)) → 1

where Aut_{B}^{0}(F_{n}) = B_{n}^{0}(R) ∩ Aut(F_{n}). As with B_{n}(R) above, we can push our use of Proposition 2.6.1 farther and get

1 → C^{0}(B_{n}(R), t_{1}) → Aut_{π}^{0}(F_{n}, t_{1}) → Out_{π}(A_{n}(R)) → 1.

The following summarizes how we can relate automorphism groups of Thompson groups to Out(A_{n}(R)) and Out(B_{n}(R)). We have been somewhat more successful with B_{n}(R) than with A_{n}(R). More information about the kernels Aut_{B}(F_{n,∞}) and Aut_{B}^{0}(F_{n}) will be given in Proposition 6.1.3.

Proposition 2.6.2. There are short exact sequences and isomorphisms

1 → Aut_{B}(F_{n,∞}) → Aut_{o}(F_{n,∞}) → Out_{o}(B_{n}(R)) → 1,
1 → Aut_{B}^{0}(F_{n}) → Aut_{π}^{0}(F_{n}) → Out_{π}(A_{n}(R)) → 1,
Out_{π}(B_{n}(R)) ≃ Aut_{π}^{0}(F_{n,∞}, t_{n−1})/C^{0}(B_{n}(R), t_{n−1}), and
Out_{π}(A_{n}(R)) ≃ Aut_{π}^{0}(F_{n}, t_{1})/C^{0}(B_{n}(R), t_{1}).

3. PL automorphisms

3.1. Inner automorhpisms.

Definition 3.1.1. We concentrate on the denominator C^{0}(B_{n}(R), t_{n−1}) in Proposition 2.6.2. The group C^{0}(B_{n}(R), t_{n−1}) is clearly isomorphic to the group B_{n}^{0}(S_{n−1}) and by Corollary 2.3.1.1 to F_{n,[0,n−1]}. The group F_{n,[0,n−1]} is a subgroup of F_{n,0} so each W ∈ F_{n,[0,n−1]} is a word in generators g_{i}, i ≥ 0. Given W ∈ F_{n,[0,n−1]} we can build an element of C^{0}(B_{n}(R), t_{n−1}) from W as the infinite composition (written as a product)

W Φ_{n} = ∏_{k=−∞}^{∞} t_{n−1}^{−k}W t_{n−1}^{k} = ∏_{k=−∞}^{∞} W σ_{n−1}^{k}.

Note that the infinite composition makes sense (and the order is irrelevant) since the supports of W σ_{n−1}^{k} and W σ_{n−1}^{j} are disjoint if k ≠ j.

Lemma 3.1.2. Let n ≥ 2.

(a) Φ_{n} : F_{n,[0,n−1]} → C^{0}(B_{n}(R), t_{n−1}) is an isomorphism.

(b) For 0 ≤ j < n − 1, W ∈ F_{n,[0,n−1]} and w = W Φ_{n}, we get w^{−1}g_{j}w = W^{−1}g_{j}W W_{1} where W_{1} = W σ_{n−1}.

Proof. (a) is clear.

Let W_{k} = W σ_{n−1}^{k}. To see (b), note first that all W_{k} commute with g_{j} when k < 0. Note next, that as a word in the generators g_{i}, W_{k} for k ≥ 2 uses only those g_{i} with i ≥ 2n − 2. These values of i exceed n − 2 by at least n, and for k ≥ 2, we get W_{k}^{−1}g_{j} = g_{j}W_{k+1}^{−1}. Since W_{k} and W_{l} commute when k ≠ l, (b) follows.

3.2. General PL automorphisms. We will show that Aut_{P L}(B_{n}(R)) has a structure similar to that of B_{n}(R), but that a larger set of slopes is used. We start with some definitions needed to accomodate the larger structure.

For an integer n ≥ 2 let ⟨⟨n⟩⟩ denote the multiplicative subgroup of Q generated by the (prime) divisors of n. We have that ⟨⟨n⟩⟩ is the group of multiplicative units of Z[\frac{1}{n}], and that n is prime iff ⟨n⟩ = ⟨⟨n⟩⟩. We let U_{n−1} denote the group of units in Z_{n−1} and we note that all divisors of n are in U_{n−1}. We let D_{n−1} denote the subgroup of U_{n−1} generated by the divisors of n. The group U_{n−1} acts on Z_{n−1} by multiplication and this action gives an isomorphism from U_{n−1} to Aut(Z_{n−1}). It seems to be a non-trivial number theoretical problem to determine the structure of D_{n−1}.

Note that ϕ_{n} : ⟨⟨n⟩⟩ → U_{n−1} is a homomorphism whose kernel contains ⟨n⟩ and whose image is D_{n−1}. Thus we get an induced epimorphism ϕ_{n} : ⟨⟨n⟩⟩/⟨n⟩ → D_{n−1}.

If A and B are groups and there is a homomorphims η : B → Aut(A), then we can form the semidirect product A ⋊ B using the action of B on A given by η. We can think of A ⋊ B as acting on A affinely in that (a, b) ∈ A ⋊ B takes g ∈ A to g^{b}a. If A is abelian, the result can be written gb + a. By sending the function x ↦ mx + a to the pair (a, m), we can identify Aff(R), the affine functions on R, with R ⋊ R^{∗} (where R^{∗} is the multiplicative group of non-zero elements of R) and we refer to Z_{n−1} ⋊ Aut(Z_{n−1}) = Z_{n−1} ⋊ U_{n−1} as Aff(Z_{n−1}), the group of affine transformations of Z_{n−1}.

We will use the following standard constructions that give new semi-direct products from A ⋊ B.

(i) If H ⊆ A is invariant under the action of G ⊆ B we get H ⋊ G ⊆ A ⋊ B with the action of H ⋊ G on H given as a restriction of the original action on A.

(ii) If N ◁ A is invariant under the action of B, then we get a surjection A ⋊ B → (A/N) ⋊ B with the action of (A/N) ⋊ B on A/N induced by the original action on A.

(iii) If K ◁B centralizes A (is in the kernel of η), then we get a surjection A⋊B → A ⋊ (B/K) with the action of A ⋊ (B/K) on A induced by the original action on A.

Recall that Π_{n} is the group of permutations of the set {0, …, n − 1}. We let Π_{n}^{0} be the subgroup of Π_{n} consisting of all elements that fix 0.

Lemma 3.2.1. Let n ≥ 2 be an integer and let f : R → R be a homeomorphism satisfying Properties 1,2, 4 and 5 of Definition 1.1.1. Then all the slopes of f are units in Z[\frac{1}{n}], and therefore elements of ⟨⟨n⟩⟩. If G is a group of homeomorphisms f : R → R satisfying Properties 1,2, 4 and 5 of Definition 1.1.1 and (3′) the slopes of f are in a single coset of ⟨⟨n⟩⟩/⟨n⟩, then:

(a) ϕ_{n} is constant on the set of slopes of any given f ∈ G,

(b) There are homomorphisms

G → Z_{n−1} ⋊ ⟨⟨n⟩⟩/⟨n⟩ → Z_{n−1} ⋊ D_{n−1} ⊆ Aff(Z_{n−1})

where the first homomorphism is given by f ↦ (0fϕ_{n}, xf′) for any real x.

(c) The composition is a homomorphism π_{n} : G → Aff(Z_{n−1}) ⊆ Π_{n−1} giving the action of any f ∈ G on the values of xϕ_{n} for x ∈ Z[\frac{1}{n}].

(d) If f ∈ G fixes 0, then fπ_{n} ∈ Π_{n−1}^{0} is the automorphism of Z_{n−1} given by multiplication by xf′ϕ_{n} for any x ∈ Z[\frac{1}{n}].

Proof. If s is the slope of one affine segment of f, and two points are given in the domain of this segment which are in Z[\frac{1}{n}] at a distance 1/n^{k} of each other, then their images will be in Z[\frac{1}{n}] at a distance s/n^{k}. This shows that s ∈ Z[\frac{1}{n}]. By Lemma 1.1.2, f^{−1} also satisfies 1,2,4,5, so 1/s ∈ Z[\frac{1}{n}] and s is a unit in Z[\frac{1}{n}].

Now suppose that all f ∈ G also satisfy 3′.

(a) Since ϕ_{n} is a ring homomorphism, and nϕ_{n} = 1, ϕ_{n} is constant on each coset in ⟨⟨n⟩⟩/⟨n⟩.

(b) An f ∈ G is made up of pieces of functions in Aff(R) so we can define [f] as the set of affine functions used by pieces of f. For f, g ∈ G we have [fg] ⊆ [f][g]. Identifying Aff(R) with R⋊R^{∗} let us note that [f] ⊆ Z[\frac{1}{n}]⋊⟨⟨n⟩⟩ which is a subgroup of R ⋊ R^{∗} by (i) above. Since Δ_{n} ⊆ Z[\frac{1}{n}] is invariant under multiplication by ⟨⟨n⟩⟩ and Z_{n−1} ≃ Z[\frac{1}{n}]/Δ_{n} is centralized by multiplication by ⟨n⟩, we get surjections

Z[\frac{1}{n}] ⋊ ⟨⟨n⟩⟩ → Z_{n−1} ⋊ ⟨⟨n⟩⟩ → Z_{n−1} ⋊ ⟨⟨n⟩⟩/⟨n⟩

from (ii) and (iii) above. We are interested in [f]^{∗} the image of [f] under the composition of these surjections.

Any two affine pieces of f that are adjacent in the graph of f give elements in [f] that agree on some element of Z[\frac{1}{n}]. For h_{1} and h_{2} in [f] and an x ∈ Z[\frac{1}{n}] on which they agree, we have xh′_{1}+0h_{1} = xh′_{2}+0h_{2} which implies that 0h_{2}−0h_{1} = x(h′_{1}−h′_{2}). Applying ϕ_{n} and using (a) gives 0h_{1}ϕ_{n} = 0h_{2}ϕ_{n}. This fact and (a) imply that [f]^{∗} is a single element of Z_{n−1} ⋊ ⟨⟨n⟩⟩/⟨n⟩. Now [fg]^{∗} ⊆ [f]^{∗}[g]^{∗} gives the first homomorphism.

The kernel of ϕ_{n} : ⟨⟨n⟩⟩/⟨n⟩ → D_{n−1} centralizes Z_{n−1} and we get the second homomorphism from (iii) above.

(c) Consider again the set [f] of affine actions on R used by pieces of an f ∈ G. Under the surjection Z[\frac{1}{n}] ⋊ ⟨⟨n⟩⟩ → Z_{n−1} ⋊ ⟨⟨n⟩⟩ induced by ϕ_{n} : Z[\frac{1}{n}] → Z_{n−1}, these actions are reduced to their actions on the values of ϕ_{n}.

(d) This is immediate.

Lemma 3.2.2. Let n ≥ 2 be an integer and let h : R → R be an orientation preserving PL homeomorphism. The following are equivalent:

(a) h normalizes A_{n}(R).

(b) h normalizes B_{n}(R).

(c) All of the following hold: (i) all the breaks of h occur at elements of Z[\frac{1}{n}] (ii) (Z[\frac{1}{n}])h = Z[\frac{1}{n}] (iii) all the slopes of h are in a single coset of ⟨⟨n⟩⟩/⟨n⟩.

Proof. (a) ⇒ (b) Follows from Proposition 1.4.2.

(b) ⇒ (c) Notice that since N(B_{n}(R)) is a group, we actually have that both h and h^{−1} normalize B_{n}(R), and it suffices to check each of the three conditions for one of h and h^{−1}. For the first condition, let x be a break point of h^{−1}, and let y = xh^{−1}. By Proposition 1.2.2 there is f ∈ B_{n}(R) that maps a neighborhood of y affinely into the domain of an affine segment of h. Since f does not break at y and h does not break at yf, we know that h^{−1}fh breaks at x. Since h^{−1}fh ∈ B_{n}(R) we get x ∈ Z[\frac{1}{n}].

We get the second condition from Lemma 1.5.2, Proposition 1.3.3 and the fact that elements of B_{n}(R) take Z[\frac{1}{n}] onto Z[\frac{1}{n}].

For the third condition, let x, y ∈ R be points where the derivative h′ is continuous. We want to show to show that xh′ ≡ yh′ (mod ⟨n⟩), i.e., (xh′)^{−1}yh′ ∈ ⟨n⟩. Since h is affine in neighborhoods of x and y, we may assume that x, y ∈ Z[\frac{1}{n}] and xϕ_{n} = yϕ_{n}. By Proposition 1.2.2 there is f ∈ B_{n}(R) such that xf = y, and moreover we may assume that f′ is continuous at x. Since h normalizes B_{n}(R), we have h^{−1}fh ∈ B_{n}(R). If we let z = xh, then the chain rule gives z(h^{−1}fh)′ = (zh^{−1}′)(zh^{−1}f′)(zh^{−1}fh′) = (xh′)^{−1}(xf′)(yh′). Since the slopes of f and h^{−1}fh are in ⟨n⟩, we get (xh′)^{−1}(yh′) ∈ ⟨n⟩, as needed.

[figure: 1; D_{n−1}; Z_{n−1} ⋊ D_{n−1}; ⟨⟨n⟩⟩/⟨n⟩; Z_{n−1} ⋊ ⟨⟨n⟩⟩/⟨n⟩; Aut_{P L}(B_{n}(R)); Aut_{P L}(B_{n}(R)); K; D_{n−1}; P L_{π}; Z_{n−1}; A_{n}(R); Q; B_{n}(R); Q; Z_{n−1}; 1; B_{n}(R)]

(a)  (b)

Figure 1. PL automorphisms

(c) ⇒ (a) Let f ∈ A_{n}(R). Since h satisfies Conditions 1,2,4 and 5 of Definition 1.1.1, then so does h^{−1}fh. Since all the slopes of h are in the same coset of ⟨⟨n⟩⟩/⟨n⟩, then the chain rule implies that the slopes of h^{−1}fh are in ⟨n⟩ and h^{−1}fh ∈ A_{n}(R).

Lemma 3.2.3. For each coset of ⟨⟨n⟩⟩/⟨n⟩ there is a homeomorphism h : R → R in Aut_{P L}(B_{n}(R)) with slopes in the given coset. Moreover, h commutes with t_{n−1} and fixes 0.

Proof. It suffices to prove the statement for cosets of the form p⟨n⟩ where p is a prime divisor of the composite n. Consider the map h_{1} : [0, n − 1] → [0, n − 1] consisting of two affine segments—one from (0, 0) to (\frac{n}{p} − 1, n − p) with slope p and the second from (\frac{n}{p} − 1, n − p) to (n − 1, n − 1) with slope p/n. Since p | n, the break points are all in Z ⊆ Z[\frac{1}{n}]. Now h = h_{1}Φ_{n} has the required properties.

Proposition 3.2.4. There are homomorphisms that commute as shown in Figure 1(a), with the kernels of those compositions with domain Aut_{P L}(B_{n}(R)) forming corresponding subgroups and consecutive quotients as shown in Figure 1(b). The group Q is defined only as the kernel of ϕ_{n} : ⟨⟨n⟩⟩/⟨n⟩ → D_{n−1}. The group K is the set of elements f ∈ Aut_{P L}(B_{n}(R)) with xf′ϕ_{n} = 1 for all x ∈ Z[\frac{1}{n}], and P L_{π} is the set of elements with fπ_{n} trivial. The diagram (b) is a lattice in that P L_{π} ∩ A_{n}(R) = B_{n}(R) and P L_{π}A_{n}(R) = K.

Proof. Two of the homomorphisms in (a) come from Lemma 3.2.1. The remaining homomorphisms and the fact that the diagram commutes are straightforward. Checking that K, P L_{π}, A_{N}(R) and B_{n}(R) are the kernels of the corresponding homomorphisms is also straightforward from the definitions. That (b) forms a lattice follows from the fact that the corresponding diagram of kernels of homomorphisms with domain Z_{n−1} ⋊ ⟨⟨n⟩⟩/⟨n⟩ forms a lattice.

Corollary 3.2.4.1. We have isomorphisms

Out_{P L}(A_{n}(R)) ≃ ⟨⟨n⟩⟩/⟨n⟩, and Out_{P L}(B_{n}(R)) ≃ Z_{n−1} ⋊ ⟨⟨n⟩⟩/⟨n⟩.

From here we can determine the torsion of Out_{P L}(A_{n}(R)) and Out_{P L}(B_{n}(R)).

Corollary 3.2.4.2. Let n ≥ 2, let k be the largest positive integer such that n is a k-th power, and let m be such that n = m^{k}. Identify Out_{P L}(A_{n}(R)) and Out_{P L}(B_{n}(R)) with the groups ⟨⟨n⟩⟩/⟨n⟩ and Z_{n−1} ⋊⟨⟨n⟩⟩/⟨n⟩ as given in Corollary 3.2.4.1.

(a) The torsion elements of Out_{P L}(A_{n}(R)) form the cyclic subgroup ⟨m⟩/⟨n⟩ of order k in ⟨⟨n⟩⟩/⟨n⟩.

(b) The torsion elements of Out_{P L}(B_{n}(R)) form the subgroup Z_{n−1} ⋊ ⟨m⟩/⟨n⟩ in Z_{n−1} ⋊ ⟨⟨n⟩⟩/⟨n⟩. The orders of the torsion elements are all the divisors of \frac{k}{d} (m^{d} − 1) where d ranges over all divisors of k.

(c) Out_{P L}(A_{n}(R)) and Out_{P L}(B_{n}(R)) have elements of infinite order if and only if n is not a prime power.

Proof. (a) Write m = p_{1}^{α_{1}} ⋯ p_{l}^{α_{l}} with p_{1}, …, p_{l} distinct primes and α_{1}, …, α_{l} > 0. Let s ∈ ⟨⟨n⟩⟩, s = p_{1}^{β_{1}} ⋯ p_{l}^{β_{l}} with β_{1}, …, β_{l} ∈ Z be such that s⟨n⟩ has finite order u. Then s^{u} = n^{v} = m^{kv} for some v ∈ Z, and gcd(u, v) = 1. It then follows that uβ_{i} = kvα_{i}, and u|kα_{i}. By the choice of k we have gcd(α_{1}, …, α_{l}) = 1 so u|k, and s = m^{\frac{k}{u} v}. Therefore the torsion elements of ⟨⟨n⟩⟩/⟨n⟩ are precisely the powers of m⟨n⟩, i.e., ⟨m⟩/⟨n⟩.

(b) For s ∈ ⟨⟨n⟩⟩ let us also use s to denote its coset s⟨n⟩ in ⟨⟨n⟩⟩/⟨n⟩. Since the action of ⟨⟨n⟩⟩/⟨n⟩ on Z_{n−1} is by multiplication, for any a ∈ Z_{n−1}, (a, s)^{u} = (a + sa + ⋯ + s^{u−1}a, s^{u}). So (a, s) has finite order if and only if s⟨n⟩ has finite order in ⟨⟨n⟩⟩/⟨n⟩. Thus the torsion elements of Z_{n−1} ⋊ ⟨⟨n⟩⟩/⟨n⟩ form the subgroup Z_{n−1} ⋊ ⟨m⟩/⟨n⟩.

For d a divisor of k,

(1, m^{d})^{\frac{k}{d}} = (\frac{m^{k} − 1}{m^{d} − 1}, m^{k}) = (\frac{n − 1}{m^{d} − 1}, 1)

so the element (1, m^{d}) has order \frac{k}{d} (m^{d} − 1).

For any 0 ≤ v < k if we let d = gcd(v, k), v′ = v/d, k′ = k/d, then for a ∈ Z_{n−1} we have

(a, m^{v})^{\frac{k}{d} (m^{d}−1)} = (a\frac{(m^{k})^{v′} − 1}{(m^{d})^{v′} − 1}, m^{v′k})^{m^{d}−1}

= (a\frac{1 + m^{k} + ⋯ + m^{k(v′−1)}}{1 + m^{d} ⋯ + m^{d(v′−1)}} · \frac{m^{k} − 1}{m^{d} − 1}, 1)^{m^{d}−1}

= (a\frac{1 + (m^{d})^{k′} + ⋯ + (m^{d})^{k′(v′−1)}}{1 + m^{d} ⋯ + m^{d(v′−1)}} (n − 1), 1) = (0, 1).

In the last line, the denominator divides the numerator since a standard exercise using the factorization of x^{r} −1 into cyclotomic polynomials [9, (2) on P. 191] shows that if P (x) = 1 + x + ⋯ x^{ν−1} and gcd(ν, k) = 1, then P (x) divides P (x^{k}).

(c) Follows from (a) and (b) and the definition of ⟨⟨n⟩⟩.

4. Relations among the automorphism groups

In this section we let the Thompson groups with elements whose slopes are in ⟨m⟩ interact with groups with elements whose slopes are in ⟨n⟩. By being careful, we keep all references to the generators g_{i} confined to those with non-negative subscript—the generators of F_{m,0} and F_{n,0}.

Because of Section 2.6 we are interested in the automorphism groups of certain pairs (G, g). We will use homomorphisms between pairs to relate their automorphism groups. For groups G and H with g ∈ G and h ∈ H, we say that θ : (G, g) → (H, h) is a homomorphism of the pairs if θ : G → H is a homomorphism and gθ = h. Recall that the definition of the automorphism group of the pair (G, g) does not require g ∈ G.

This section studies monomorphisms between the pairs (F_{n,∞}, t_{n−1}) for various values of n and uses this information to develop two techniques, lifting and rotating, that create “new” automorphisms of B_{n}(R) and A_{n}(R) from “old” ones. We also study how lifting and rotating interact with the homomorphisms π_{n} to show that the π_{n} are onto.

4.1. Lifts. We are interested in automorphisms of the pair (F_{n,∞}, t_{n−1}). We will obtain interesting automorphisms of the pair (F_{n,∞}, t_{n−1}) from automorphisms of the pair (F_{m,∞}, t_{m−1}), m < n, by the process that we will call lifting whose details are given in Theorem 4.1.6. We do this by showing that there is a monomorphism from (F_{m,∞}, t_{m−1}) to (F_{n,∞}, t_{n−1}) that is “extensible” in that every automorphism of the pair (F_{m,∞}, t_{m−1}) extends to one of the pair (F_{n,∞}, t_{n−1}).

Definition 4.1.1. The proof of the next lemma, and several others after, will use the following function. Let 2 ≤ m < n be integers. For every j ∈ Z there are unique 0 ≤ i ≤ (m−2) and k ∈ Z for which j = i+k(m−1). If we set jζ_{m,n} = i+k(n−1) with i, j and k as in the previous sentence, then we have a well defined, strictly increasing function from Z to Z that satisfies satisfies (j+m−1)ζ_{m,n} = jζ_{m,n}+n−1, and that “commutes” with ϕ in that (jζ_{m,n})ϕ_{n} = jϕ_{m} for any j ∈ Z.

Proposition 4.1.2. For integers 2 ≤ m < n, there is a unique monomorphism of group pairs τ_{m,n} : (F_{m,∞}, t_{m−1}) → (F_{n,∞}, t_{n−1}) so that (g_{m,i})τ_{m,n} = g_{n,i} for 0 ≤ i ≤ (m − 2). For this monomorphism, (g_{m,j})τ_{m,n} = g_{n,jζ_{m,n}} for all j.

Proof. With j equal to the unique expression i+k(m−1) with 0 ≤ i ≤ (m−2) and k ∈ Z, we get that g_{m,j} is equal to a unique t_{m−1}^{−k}(g_{m,i})t_{m−1}^{k} for 0 ≤ i ≤ (m − 2) and k ∈ Z. From the requirement that t_{m−1}τ_{m,n} = t_{n−1}, we get

(g_{m,j})τ_{m,n} = (t_{m−1}^{−k}g_{m,i}t_{m−1}^{k})τ_{m,n} = t_{n−1}^{−k}g_{n,i}t_{n−1}^{k} = g_{n,i+k(n−1)} = g_{n,jζ_{m,n}}

and the function τ_{m,n} is uniquely determined on its generators. There is thus at most one monomorphism with the properties given.

The defining relations of F_{m,∞} take the form g_{m,i}^{−1}g_{m,j}g_{m,i} = (g_{m,j+n−1}) whenever i < j. Now

(g_{m,i}^{−1}τ_{m,n})(g_{m,j}τ_{m,n})(g_{m,i}τ_{m,n}) = g_{n,iζ_{m,n}}^{−1}g_{n,jζ_{m,n}}g_{n,iζ_{m,n}}
= g_{n,jζ_{m,n}+n−1}
= g_{n,(j+m−1)ζ_{m,n}}
= (g_{m,j+m−1})τ_{m,n}

where the second equality holds because iζ_{m,n} < jζ_{m,n}. Thus τ_{m,n} preserves the relations of F_{m,∞} and extends to a homomorphism. It is a monomorphism by Proposition 2.1.5 since the elements g_{n,i} do not commute.

The next few lemmas will be needed to show that automorphisms of the image of the monomorphism of the previous lemma extend to automorphisms of the larger group. We refer to Definition 2.2.1 for what it means for a subdivision of R to be supported in a subset A ⊆ R.

Lemma 4.1.3. For n ≥ 2, let P = g_{i_{1}}g_{i_{2}} ⋯ g_{i_{s}} be a positive word in semi-normal form in the generators of F_{n,0} and let k < l be integers. Then the subdivision D associated with P is supported in [k, l] if and only if k ≤ i_{1} and for all 1 ≤ j ≤ s we have i_{j} < l + (n − 1)(j − 1).

Note the shift in role of j from subscript to value in the last inequality.

Proof. The interval subdivided by g_{i_{1}} is the interval [i_{1}, i_{1} + 1] so the conditions on i_{1} are necessary and sufficient as far as i_{1} goes. We need to show that the second condition is necessary and sufficient to guarantee that the interval subdivided by each remaining generator is contained somewhere within [k, l]. Since P is in seminormal form, i_{j} ≥ i_{j−1} for each 2 ≤ j ≤ s. Let [k_{j}, k_{j} + 1], k_{j} ∈ Z, be the standard interval containing the interval subdivided by g_{i_{j}}. It is clear from the numbering scheme of Definition 2.2.3 that the interval subdivided by g_{i_{j}} is either contained in [k_{j−1}, k_{j−1} + 1] and k_{j} = k_{j−1}, or it is some [k_{j}, k_{j} + 1] with k_{j} ∈ Z and k_{j} > k_{j−1}. Thus the interval subdivided by g_{i_{j}} cannot be to the left of k.

In the numbering scheme of Definition 2.2.3, the interval [k_{j}, k_{j} + 1] is assigned the number k_{j} before any subdivisions are done and this number increases by (n−1) each time an interval to the left of [k_{j}, k_{j} + 1] is subdivided. If k_{j} > k_{j−1}, then all generators in P before g_{i_{j}} are associated to subdivisions of intervals to the left of [k_{j}, k_{j}+1] and the number of [k_{j}, k_{j}+1] is k_{j}+(n−1)(j−1) = i_{j} when the subdivision associated to g_{i_{j}} is done. If k_{j} = k_{j−1}, then when the subdivision associated to g_{i_{j}} is done, the number of [k_{j−1} + 1, k_{j−1} + 2] is k_{j−1} + 1 + (n − 1)(j − 1) > i_{j} and we get i_{j} ≤ k_{j} + (n − 1)(j − 1). If all k_{j} < l, then all i_{j} < l + (n − 1)(j − 1). For the converse, assume all i_{j} < l + (n − 1)(j − 1). Clearly k_{1} = i_{1} < l. If k_{j} = k_{j−1}, we are done by induction, and if k_{j} > k_{j−1}, then k_{j} = i_{j} − (n − 1)(j − 1) < l.

Lemma 4.1.4. For 2 ≤ m < n, let P = g_{i_{1}}g_{i_{2}} ⋯ g_{i_{s}} be a positive word in seminormal form in the generators of F_{m,0} and let k < l be integers. Assume the subdivision D associated with P is supported in [k, l]. Then the subdivision associated with P τ_{m,n} is supported in [k′, l′] where k′ = kζ_{m,n} and l′ = (l − 1)ζ_{m,n} + 1.

Proof. From Lemma 4.1.3, we have k ≤ i_{1} so k′ = kζ_{m,n} ≤ i_{1}ζ_{m,n} and k′ satisfies the only inequality from Lemma 4.1.3 that it needs to satisfy. Now i_{j} < l + (m − 1)(j − 1) can be rewritten i_{j} ≤ (l − 1) + (m − 1)(j − 1) which implies i_{j}ζ_{m,n} ≤ (l − 1)ζ_{m,n} + (n − 1)(j − 1) and we have i_{j}ζ_{m,n} < (l − 1)ζ_{m,n} + 1 + (n − 1)(j − 1) and we are done.

In the next lemma, the support of a homeomorphism refers to its set of non-fixed points.

Lemma 4.1.5. For 2 ≤ m < n, let f be in F_{m,0}, let k < l be integers, let k′ = kζ_{m,n} and let l′ = (l − 1)ζ_{m,n} + 1. (a) If f has support in [k, l], then fτ_{m,n} has support in [k′, l′]. (b) If f is the identity to the left of k and is translation by (m−1) to the right of l, then fτ_{m,n} is the identity to the left of k′ and translation by (n−1) to the right of l′.

Proof. Let f = P N^{−1} be in semi-normal form where P and N are positive words. There are allowable subdivisions D and E associated with P and N respectively. We can assume that D subdivides no interval [s, s + 1], s ∈ Z, that is outside [k, l] since such a subdivision would have to be matched exactly by E on [s, s+1] in case (a) or [s + m − 1, s + m] in case (b) and the two matching subdivisions could be removed. Similarly, E can be assumed to subdivide no such interval outside [k, l] in case (a) or outside [k, l + m − 1] in case (b). We also note the the number of generators in P equals the number in N in case (a) and exceeds the number in N by one in case (b).

By Lemma 4.1.4, P′ = P τ_{m,n} and N′ = Nτ_{m,n} are supported in [k′, l′] and [k′, l′] respectively in case (a) or [k′, l′] and [k′, l′ + n − 1] respectively in case (b). Further the number of generators in P′ equals the number of generators of N′ in case (a) and exceeds the number in N′ by one in case (b). Since fτ_{m,n} = P′(N′)^{−1} is the identity near −∞, the conclusions in (a) and (b) follow from the information gathered.

Theorem 4.1.6. Let 2 ≤ m < n be integers. Let τ_{m,n} be as in Proposition 4.1.2. Then there is a monomorphism

Θ_{m,n} : Aut^{0}(F_{m,∞}, t_{m−1}) → Aut^{0}(F_{n,∞}, t_{n−1})

so that if θ′ = θΘ_{m,n}, then (a) g_{n,i}θ′ = (g_{m,i}θ)τ_{m,n} for 0 ≤ i ≤ (m − 2) and g_{n,j}θ′ = g_{n,j} for (m − 1) ≤ j ≤ (n − 2), and (b) (fτ_{m,n})θ′ = (fθ)τ_{m,n} for all f ∈ F_{m,∞}.

Proof. Note that (a) specifies θ′ on the generators g_{n,i} of F_{n,∞} for 0 ≤ i ≤ (n − 2). If θ′ exists, then it is sufficiently specified by (a) since it is required to fix t_{n−1} and F_{n,∞} is generated by {t_{n−1}, g_{n,0}, …, g_{n,n−2}}. The specifications in (a) for 0 ≤ i ≤ (m−2) imply that the equality in (b) holds when f = g_{m,i}, 0 ≤ i ≤ (m−2), and when f = t_{m−1}. These are the generators of F_{m,∞} and (b) will hold when it is shown that θ′ is a homomorphism.

We have to show two things. For a given θ ∈ Aut^{0}(F_{m,∞}, t_{m−1}), we have to show that θ′ = θΘ_{m,n} is an automorphism of the pair (F_{n,∞}, t_{n−1}). We also have to show that Θ_{m,n} is an injective homomorhism. That Θ_{m,n} is a homomorphism is clear from the definition. That Θ_{m,n} is a monomorphism follows from the fact that if θ does not fix some g_{m,i}, then (g_{m,i}θ)τ_{m,n} ≠ g_{n,i} since τ_{m,n} is a monomorphism.

To show that θ′ is an automorphism, we have to show that it is an invertible homomorphism. To show that it is a homomorphism, we have to show that the images of the generators of F_{n,∞} satisfy the relations of F_{n,∞}.

Since θ ∈ Aut^{0}(F_{m,∞}, t_{m−1}), we know that θ is realized as conjugation by a self homeomorphism of R that commutes with t_{m−1} and that fixes 0. We will abuse notation and use θ to denote this self homeomorphism of R. Thus 0θ = 0 and (m − 1)θ = (m − 1).

Take an i with 0 ≤ i ≤ (m − 2). The generator g_{m,i} is the identity to the left of i and is equal to t_{m−1} to the right of i + 1. Thus θ^{−1}g_{m,i}θ is the identity to the left of iθ and is equal to t_{m−1} to the right of (i + 1)θ. However, 0 ≤ i ≤ (m − 2) gives 0 ≤ i < i + 1 ≤ (m − 1) which leads to 0 ≤ iθ < (i + 1)θ ≤ (m − 1) and θ^{−1}g_{m,i}θ is the identity to the left of 0 and equals t_{m−1} to the right of m − 1. It follows from Lemma 4.1.5 that (g_{n,i})θ′ is the identity to the left of 0 and equals t_{n−1} to the right of m − 1.

We have g_{n,i+n−1}θ′ = (t_{n−1}^{−1}g_{n,i}t_{n−1})θ′ = t_{n−1}^{−1}(g_{n,i}θ′)t_{n−1} since this is how θ′ is defined.

Checking that (g_{n,i}^{−1}θ′)(g_{n,j}θ′)(g_{n,i}θ′) = (g_{n,j+n−1}θ′) = t_{n−1}^{−1}(g_{n,j}θ′)t_{n−1} for all i < j is straightforward by considering separately the various cases where g_{n,i} and g_{n,j} are or are not in the image of τ_{m,n}.

This establishes that θ′ is an endomorphism of F_{n,∞}. Since Θ_{m,n} preserves compositions and inverses, θ′ is invertible.

Corollary 4.1.6.1. Given 2 ≤ m < n, f ∈ F_{m,∞} and α and β in Aut^{0}(f_{m,∞}, t_{m−1}) with fα = fβ, then (fτ_{m,n})(αΘ_{m,n}) = (fτ_{m,n})(βΘ_{m,n}).

4.2. Rotations.

Lemma 4.2.1. For n ≥ 2, let α ∈ Aut^{0}(F_{n,∞}, t_{n−1}) and let h : R → R be the homeomorphism fixing 0 and commuting with t_{n−1} that realizes α. Let j ∈ Z[\frac{1}{n}] and let r = jh. Then k = t_{j}ht_{r}^{−1} is a homeomorphism of R that fixes 0 and commutes with t_{n−1}. Therefore it realizes a β ∈ Aut^{0}(F_{n,∞}, t_{n−1}).

Proof. Since t_{n−1} commutes with h, t_{j} and t_{r}, it commutes with k. Now, 0k = 0t_{j}ht_{r}^{−1} = jht_{r}^{−1} = rt_{r}^{−1} = 0.

In the setting of the lemma above we call k (resp. β) the j-step rotation of h (resp. α). While the j-step rotation is a function from Aut^{0}(F_{n,∞}, t_{n−1}) to itself, it is not in general a homomorphism. An elementary calculation shows that the j′-step rotation of the j-step rotation of h is the (j + j′)-step rotation of h. We are mainly interested in the case when j is an integer. Note that the (n − 1)-step rotation of h is h.

Proposition 4.2.2. For n ≥ 2, let α ∈ Aut^{0}(F_{n,∞}, t_{n−1}) and let h : R → R be the homeomorphism fixing 0 and commuting with t_{n−1} that realizes α. Let j be an integer, and k be the j-step rotation of h.

(a) h normalizes A_{n}(R) if and only if k does.

(b) If h commutes with t_{1} then k = h. Moreover, h commutes with t_{1} if and only if h equals its one-step rotation.

(c) kπ_{n} = ρ_{jϕ_{n}}(hπ_{n})ρ_{(jϕ_{n})(hπ_{n})}^{−1} where ρ_{i} denotes addition of i (mod n−1) in the set {0, 1, …, n − 2}. In particular, if h fixes the residue class of j then kπ_{n} and hπ_{n} are conjugate.

(d) h fixes residue classes (i.e., hπ_{n} ≡ 1) if and only if k does.

Proof. Since h is the (n − 1 − j)-step rotation of k, in parts (a) and (d) it suffices to prove one direction.

(a) This follows because translations by elements of Z[\frac{1}{n}] normalize A_{n}(R).

(b) If h commutes with t_{1} then r = jh = 0t_{1}^{j}h = 0ht_{1}^{j} = 0t_{1}^{j} = j so k = t_{j}ht_{r}^{−1} = t_{1}^{j}ht_{1}^{−j} = h. For the converse, note that if t_{1}ht_{1h}^{−1} = h, then t_{1}^{n−1}ht_{1h}^{−n+1} = h. However, h commutes with t_{1}^{n−1} and we get that (t_{1}t_{1h}^{−1})^{n−1} is the identity. Translations form a torsion free group, so t_{1} = t_{1h}. Now we know that h commutes with t_{1}.

(c) From Proposition 1.5.4 we have that (jh)ϕ_{n} = (jϕ_{n})(hπ_{n}), so

kπ_{n} = (t_{j}π_{n})(hπ_{n})(t_{jh}π_{n})^{−1} = ρ_{jϕ_{n}}(hπ_{n})ρ_{(jh)ϕ_{n}}^{−1} = ρ_{jϕ_{n}}(hπ_{n})ρ_{(jϕ_{n})(hπ_{n})}^{−1}.

(d) Follows at once from (c).

4.3. Symmetric lifts. We introduce special lifts that will create automorphisms of A_{n} from automorphisms of A_{2} whenever n ≥ 2. See Theorem 6.1.5.

Theorem 4.3.1. Let 2 ≤ m < n be integers for which (m − 1) | (n − 1) and let τ_{m,n} : (F_{m,∞}, t_{1}) → (F_{n,∞}, t_{n−1}) be the monomorphism of Proposition 4.1.2. Then there is a monomorphism

Λ_{m,n} : Aut^{0}(F_{m,∞}, t_{m−1}) → Aut^{0}(F_{n,∞}, t_{m−1})

(note the mix of m and n in the right hand group) taking θ ∈ Aut^{0}(F_{m,∞}, t_{m−1}) to θ′ ∈ Aut^{0}(F_{n,∞}, t_{m−1}) so that g_{n,i}θ′ = (g_{m,i}θ)τ_{m,n} for 0 ≤ i ≤ m − 2.

Proof. The proof is similar to that of Theorem 4.1.6. With θ′ defined on g_{n,i}, 0 ≤ i ≤ m − 2, we can set g_{n,k}θ′ = (g_{n,j}θ′)σ_{m−1}^{l} where 0 ≤ j ≤ m − 2 and k = j + l(m − 1). Thus θ′ is sufficiently specified. The main tasks are to show that θ′ is an automorphism and that Λ_{m,n} is an injective homomorphism. An argument identical to that in the proof of Theorem 4.1.6 shows that the g_{n,k}θ′ satisfy the relations of F_{n,∞}. That θ′ is an automorphism follows as it does in Theorem 4.1.6. By the way θ′ is defined, it commutes with t_{m−1}, and it is thus an automorphism of (F_{n,∞}, t_{m−1}). That Λ_{m,n} is a monomorphism is argued as it is for Θ_{m,n} in Theorem 4.1.6.

It might be surmised that if 2 < m < n and (m − 1) | (n − 1), then a θ ∈ Aut^{0}(F_{m,∞}, t_{m−1}) that is also a normalizer of A_{m}(R) should lift to a θ′ = θΛ_{m,n} ∈ Aut^{0}(F_{n,∞}, t_{m−1}) that is also a normalizer of A_{n}(R). We will see in Theorem 5.4.8 that this is not always the case.

4.4. Lifting permutations. Let n ≥ 2. For w ∈ F_{n,−∞}, w ≠ 1, the leftmost break of w is the largest a ∈ R such that w is the identity on the interval (−∞, a]. Also recall that Π_{n} is the permutation group on {0, …, n − 1} and that Π_{n}^{0} is the subgroup of elements that fix 0.

Lemma 4.4.1. For n ≥ 2, let w ∈ F_{n,−∞}, w ≠ 1, and let a be the leftmost break of w.

(a) If h is an orientation preserving normalizer of F_{n,∞}, then h^{−1}wh ∈ F_{n,−∞} and the leftmost break of h^{−1}wh is ah.

(b) Let w = P N^{−1} be a semi-normal form of w. Let v = g_{i_{0}}g_{i_{1}} … g_{i_{k−1}} be the largest common prefix of P and N as positive words in the generators. Let g_{i_{k}} be the lowest indexed generator in the remaining suffixes. Then aϕ_{n} = i_{k}ϕ_{n}.

Proof. Item (a) follows because conjugation takes fixed points to fixed points. To work on (b), we note that P′ = v^{−1}P and N′ = v^{−1}N start with different generators, and that the lower subscript of the two starting generators is i_{k}. This implies that the semi-normal form P′(N′)^{−1} has its leftmost break at i_{k}. The leftmost break of P N^{−1} = vP′(N′)^{−1}v^{−1} is at (i_{k})v^{−1}. Since v is a word in elements with fixed points, we have that v is in B_{n}(R) and that v preserves values of ϕ_{n}.

Proposition 4.4.2. Let 2 ≤ m < n be integers. Let τ_{m,n} and Θ_{m,n} be as in Theorem 4.1.6. Let θ ∈ Aut^{0}(F_{m,∞}, t_{m−1}) and θ′ = θΘ_{m,n} ∈ Aut^{0}(F_{n,∞}, t_{n−1}). We will abuse notation and denote by θ (resp. θ′) the self homeomorphism of R that fixes 0, commutes with t_{m−1} (resp. t_{n−1}) and realizes the automorphism θ (resp. θ′). Then the permutation θ′π_{n} is an extension to Π_{n−1}^{0} of θπ_{m} in the sense that for 1 ≤ i ≤ m − 2, i(θ′π_{n}) = i(θπ_{m}) and for m − 1 ≤ i ≤ n − 2, i(θ′π_{n}) = i.

Proof. Since θ′ carries the fixed point set of g_{n,i} to that of g_{n,i}θ′, we have that iθ′ is the leftmost break of (g_{n,i})θ′. By the way Θ_{m,n} is defined for m − 1 ≤ i ≤ n − 2, (g_{n,i})θ′ = g_{n,i} so its leftmost break is i and i(θ′π_{n}) = i in these cases. For 1 ≤ i ≤ m − 2, write (g_{m,i})θ = P N^{−1} in semi-normal form, and let v = g_{i_{0}}g_{i_{1}} … g_{i_{k−1}} be the largest common prefix of P and N as positive words in the generators. Let g_{i_{k}} be the lowest indexed generator in the remaining suffixes. Now (P τ_{m,n})(Nτ_{m,n})^{−1} is a semi-normal form for (g_{n,i})θ′ where vτ_{m,n} is the largest common prefix of P τ_{m,n} and Nτ_{m,n} and g_{m,i_{k}}τ_{m,n} = g_{n,i_{k}ζ_{m,n}} is the lowest indexed generator in the remaining suffixes. Now

i(θ′π_{n}) = (i_{k}ζ_{m,n})ϕ_{n} = i_{k}ϕ_{m} = i(θπ_{m})

by the previous lemma and the remarks in Definition 4.1.1.

Theorem 4.4.3. For any n ≥ 3, the homomorphism π_{n} : Aut(F_{n,∞}, t_{n−1}) → Π_{n−1} is surjective. Further, the restriction π_{n} : Aut^{0}(F_{n,∞}, t_{n−1}) → Π_{n−1}^{0} is surjective.

Proof. We start with the restricted homomorphism. There is nothing to show for n = 3 since Π_{2}^{0} is trivial. For n = 4, Lemma 3.2.3 gives a self homeomorphism h of R that fixes 0 and commutes with t_{3} with all its slopes of the form 2 · 4^{k}, k ∈ Z. Consider α ∈ Aut^{0}(F_{4,∞}, t_{3}) the automorphism realized by h. By Lemma 3.2.1(d), απ_{4} is multiplication by 2 (mod 3), and therefore it is the transposition (1 2). For n > 4, let α′ = αΘ_{4,n} be the lift of α from 4 to n. From Proposition 4.4.2 we have that α′π_{n} = (1 2). From Proposition 4.2.2(c), the permutation of the 1-step rotation of α′ is

ρ_{1}(α′π_{n})ρ_{2}^{−1} =ρ_{1}(α′π_{n})ρ_{n−3}
= (\begin{matrix}0 & 1 & 2 & … & n − 3 & n − 2 \\ 1 & 2 & 3 & … & n − 2 & 0 \end{matrix}) · (\begin{matrix}0 & 1 & 2 & 3 & … & n − 2 \\ 0 & 2 & 1 & 3 & … & n − 2 \end{matrix}) ·
(\begin{matrix}0 & 1 & 2 & 3 & … & n − 2 \\ n − 3 & n − 2 & 0 & 1 & … & n − 4 \end{matrix})
= (\begin{matrix}0 & 1 & 2 & 3 & … & n − 2 \\ 0 & n − 2 & 1 & 2 & … & n − 3 \end{matrix})

which is a cyclic rotation of the elements {1, 2, …, n − 2}. This rotation and the transposition (1 2) generate the whole group Π_{n−1}^{0}.

For the unrestricted homomorphism, σ_{1} realized as conjugation by t_{1} is in Aut(F_{n,∞}, t_{n−1}) and we get that the cyclic rotation of {0, …, n − 1} is in the image. For n = 3, this is the only non-trivial element in Π_{2}. For n > 3, this rotation and the transposition (1 2) found above generate all of Π_{n−1}. This completes the proof.

5. Restrictions on automorphisms

In this section we extract geometric information from the circle, and transfer that information to the line to get “negative” results about automorphisms—that certain automorphisms are not PL and that certain automorphisms of B_{n}(R) are not automorphisms of A_{n}(R).

5.1. Communication between line and circle. Because of Section 2.6, we are interested in Aut^{0}(F_{n,∞}, t_{n−1}). If α ∈ Aut^{0}(F_{n,∞}, t_{n−1}), then α is realized by conjugation by some homeomorphism h : R → R that commutes with t_{n−1} and fixes 0. This tells us that [0, n − 1]h = [0, n − 1]. From the definition

xg_{i} = \begin{cases}x & x < i, \\ n(x − i) + i & i ≤ x ≤ i + 1, \\ x + n − 1 & x > i + 1. \end{cases}

we know that g_{i} is the identity on (−∞, i], is t_{n−1} on [i + 1, ∞) and is μ̃_{n}t_{n−1}^{−i} on [i, i + 1] where μ̃_{n} : R → R is defined by xμ̃_{n} = nx and is the lift of μ_{n} : S_{n−1} → S_{n−1} that fixes 0. Now

h^{−1}g_{i}h = \begin{cases}identity & on (−∞, i]h, \\ (h^{−1}μ̃_{n}h)t_{n−1}^{−i} & on [i, i + 1]h \\ t_{n−1} & on [i + 1, ∞)h. \end{cases}

since h commutes with t_{n−1}. Recall S_{n−1} = R/(n−1)Z, and note that h determines a well defined self homeomorphism h_{S} of S_{n−1} with

(x + (n − 1)Z)h_{S} = (xh) + (n − 1)Z

and that h is the lift of h_{S} that fixes 0. If we refer to the restriction of g_{i} to [i, i + 1] as the “active” part of g_{i} and the restriction of h^{−1}g_{i}h to [i, i + 1]h as the “active” part of h^{−1}g_{i}h, then we have that the active part of g_{i} is a restriction to [i, i + 1] of that lift of μ_{n} that fixes i and the active part of h^{−1}g_{i}h is a restriction to [i, i + 1]h of that lift of h_{S}^{−1}μ_{n}h_{S} that fixes ih. For 0 ≤ i < n−1, we have ih ∈ [0, n−1). The active parts of the h^{−1}g_{i}h determine the functions h^{−1}g_{i}h. Since {t_{n−1}, g_{0} … g_{n−2}} is a generating set for F_{n,∞}, the functions h^{−1}g_{i}h, 0 ≤ i < n − 1, determine the action of α ∈ Aut^{0}(F_{n,∞}, t_{n−1}). Lastly, we note that the integers in S_{n−1} are the fixed points of μ_{n} and their images under h_{S} are the fixed points of h_{S}^{−1}μ_{n}h_{S}. All of these observations lead to the next definition and lemma.

Let α ∈ Aut^{0}(F_{n,∞}, t_{n−1}) be realized as conjugation by h : R → R, and let h_{S} : S_{n−1} → S_{n−1} be determined by h. We call the degree n function h_{S}^{−1}μ_{n}h_{S} the characteristic function χ_{α} of α. The remarks above have proven the following.

Lemma 5.1.1. For n ≥ 2, an α ∈ Aut^{0}(F_{n,∞}, t_{n−1}) is determined by its characteristic function χ_{α}. Further, for 0 ≤ i < n − 1, the active part of g_{i}α is a lift of the restriction of χ_{α} to the closed interval between the two consecutive fixed points ih and (i + 1)h of χ_{α}.

5.2. The calculus of break values and a criterion for piecewise linearity. If f is PL, then we define the break value xf^{b} of f at x to be log(xf′_{+}) − log(xf′_{−}) where xf′_{+} is the derivative from the right of f at x and xf′_{−} is the derivative from the left of f at x. We have xf^{b} = 0 if and only if f is affine on some neighborhood of x.

Lemma 5.2.1. For PL functions f, g and h with h invertible, we have

\begin{aligned}x(fg)^{b} & = xf^{b} + (xf)g^{b}, \\ (xh)(h^{−1})^{b} & = −xh^{b}, and \\ (xh)(h^{−1}gh)^{b} & = −xh^{b} + xg^{b} + (xg)h^{b}. \end{aligned}

Proof. The first equation is an additive chain rule and follows from the definition. The second equation follows by applying the chain rule to x(hh^{−1})^{b} and the third follows from the first two.

Lemma 5.2.2. For n ≥ 2, if h is a normalizer of B_{n}(S_{n−1}), then h^{−1}μ_{n}h has slope n on some neighborhood of all its fixed points and thus has break value 0 at all its fixed points.

Proof. Conjugation by h induces an isomorphism from the group of germs of B_{n}(S_{n−1}) with a fixed point x ∈ Z[\frac{1}{n}] to the group of germs with the fixed point xh. The groups involved are isomorphic to Z × Z with generators “slope n to the right” and “slope n to the left.” Conjugation by h preserves “to the right” and “to the left” in that behavior of the conjugate to one side is determined by behavior of the germ being conjugated on an appropriate side (depending on whether h preserves orientation or not). The generator to one side is taken to the generator on the corresponding side or its inverse. But conjugation takes a repelling fixed point to a repelling fixed point. Thus “slope n to the left” must be taken to “slope n to the left” as opposed to “slope \frac{1}{n} to the left.” A similar statement applies to the right. Since μ_{n} has slope n on both sides of all its fixed points, the result follows.

The above lemma and Lemma 5.1.1 give the following.

Corollary 5.2.2.1. For n ≥ 2, if h is a normalizer of F_{n,∞} that commutes with t_{n−1}, then h^{−1}g_{i}h has slope n on some [i, i+ϵ_{1})h and on some (i+1−ϵ_{2}, i+1]h.

Lemma 5.2.3. For n ≥ 2, if h is a non-trivial PL normalizer of B_{n}(S_{n−1}) that fixes 0, then h^{−1}μ_{n}h has a non-zero break value in each open interval (i, i + 1)h.

Proof. Since h is not the identity on S_{n−1}, it has a non-zero break value at some point x. Now (i, i + 1) hits each point of S_{n−1} at least once under μ_{n}. Let y be the point in (i, i + 1) closest to i so that yμ_{n} is a point at which h has a non-zero break value. Note that y is y′μ_{n} for a y′ ∈ (i, y) where the distance from i to y is n times the distance from i to y′. Thus y′μ_{n} = y is a point at which h has zero break value. Now μ_{n} has zero break value at all points, so (yh)(h^{−1}μ_{n}h)^{b} = (yμ_{n})h^{b} which is not zero.

5.3. The unbent generator proposition and locally inner automorphisms. All results in the rest of this section hinge on the next proposition and its consequences. This proposition will allow us to derive information by studying the effect of an automorphism on a single generator. Let α be in Aut^{0}(F_{n,∞}, t_{n−1}). We say that α has an unbent generator at i if g_{i}^{α} is affine on the interval [i, i + 1]h, where h is the self homeomorphism of R that realizes α.

Proposition 5.3.1 (Unbent Generator). For n ≥ 2, let α ∈ Aut^{0}(F_{n,∞}, t_{n−1}) be piecewise linear. If for some 0 ≤ i < n − 1, α has an unbent generator at i, then α is the identity.

Proof. This follows from Lemma 5.1.1 and the contrapositive of Lemma 5.2.3.

The following definitions will help us concentrate on the effect that automorphisms have on single generators.

Definition 5.3.2. Let α be in Aut^{0}(F_{n,∞}, t_{n−1}), n ≥ 2.

(a) We say that α is inner if there is w ∈ C^{0}(B_{n}(R), t_{n−1}) such that g_{i}α = w^{−1}g_{i}w for all i.

(b) We say that α is inner at i (via w) if there is w ∈ C^{0}(B_{n}(R), t_{n−1}) such that g_{i}α = w^{−1}g_{i}w.

(d) We say that α is locally inner if it is inner at every i ∈ Z.

Note that if α is inner at i then it is inner at i + k(n − 1) for every k, and the same w works for all k. Specifically

(g_{i+k(n−1)})α = (g_{i})σ_{n−1}^{k}α = (g_{i})ασ_{n−1}^{k} = (w^{−1}g_{i}w)σ_{n−1}^{k} = w^{−1}g_{i+k(n−1)}w.

Note also that α is locally inner if it is inner at every i ∈ {0, …, n − 2}. Thus if α is locally inner, then its action is completely determined by an (n − 1)-tuple (w_{0}, …, w_{n−2}) of elements of C^{0}(B_{n}(R), t_{n−1}), so that g_{i}α = w_{i}^{−1}g_{i}w_{i} for i ∈ {0, …, n − 2}. In turn, by Lemma 3.1.2(a) these w_{i}’s are determined by an (n − 1)- tuple (W_{0}, …, W_{n−2}) of elements of F_{n,[0,n−1]} with w_{i} = W_{i}Φ_{n}. Clearly, not every such (n − 1)-tuple determines a locally inner automorphism.

We now define Aut_{I_{i}}^{0}(F_{n,∞}, t_{n−1}) to be the set of elements in Aut^{0}(F_{n,∞}, t_{n−1}) that are inner at i. We will see shortly that this is a group. We let Aut_{g_{i}}^{0}(F_{n,∞}, t_{n−1}) be those elements in Aut^{0}(F_{n,∞}, t_{n−1}) that fix g_{i}. We have Aut_{g_{i}}^{0}(F_{n,∞}, t_{n−1}) ⊆ Aut_{I_{i}}^{0}(F_{n,∞}, t_{n−1}) since an automorphism fixing g_{i} is inner at i via 1. We next let Aut_{LI}^{0}(F_{n,∞}, t_{n−1}) be the set of locally inner automorphisms in Aut^{0}(F_{n,∞}, t_{n−1}). This is the intersection of the Aut_{I_{i}}^{0}(F_{n,∞}, t_{n−1}) as i ranges over {0, …, n − 2}.

Lemma 5.3.3. Let 0 ≤ i < n − 1 where n ≥ 2.

(a) If α ∈ Aut^{0}(F_{n,∞}, t_{n−1}) is inner at i via both w and w′ in C^{0}(B_{n}(R), t_{n−1}), then w = w′.

(b) Let α ∈ Aut^{0}(F_{n,∞}, t_{n−1}) be inner at i. Then either α is inner or α is not PL.

(c) Aut_{I_{i}}^{0}(F_{n,∞}, t_{n−1}) is a group and has C^{0}(B_{n}(R), t_{n−1}) as a normal subgroup with complement Aut_{g_{i}}^{0}(F_{n,∞}, t_{n−1}). Any element in Aut_{I_{i}}^{0}(F_{n,∞}, t_{n−1}) and outside C^{0}(B_{n}(R), t_{n−1}) is not PL.

(d) ⟨Aut_{g_{i}}^{0}(F_{n,∞}, t_{n−1})⟩π_{n} = ⟨Aut_{I_{i}}^{0}(F_{n,∞}, t_{n−1})⟩π_{n} = Π_{n}^{0,i,i+1} the group of permutations that fix {0, i, i + 1}.

(e) Aut_{LI}^{0}(F_{n,∞}, t_{n−1}) is a subgroup of Aut_{π}^{0}(F_{n,∞}, t_{n−1}). In general (for example, when n is not a prime power) this subgroup is proper. Any element in Aut_{LI}^{0}(F_{n,∞}, t_{n−1}) and not in C^{0}(B_{n}(R), t_{n−1}) is not PL.

Proof. (a) If w ≠ w′, then conjugation by the PL homeomorphism w′w^{−1} is a nontrivial element of Aut^{0}(F_{n,∞}, t_{n−1}) that fixes g_{i}. This violates Proposition 5.3.1.

(b) If α is inner at i via w, then α′ ∈ Aut^{0}(F_{n,∞}, t_{n−1}) obtained by composing α with conjugation by w^{−1} fixes g_{i}. By Proposition 5.3.1, α′ is either trivial or not PL.

(c) To show closure under inversion, let α ∈ Aut^{0}(F_{n,∞}, t_{n−1}) be inner at i via the element w ∈ C^{0}(B_{n}(R), t_{n−1}). From g_{i}α = w^{−1}g_{i}w, we get that g_{i} = w(g_{i}α)w^{−1} and g_{i}α^{−1} = (wα^{−1})g_{i}(w^{−1}α^{−1}) so α^{−1} is inner at i via (w^{−1}α^{−1}). To show closure under composition, let α_{1}, α_{2} ∈ Aut^{0}(F_{n,∞}, t_{n−1}) be inner at i via w_{1}, w_{2} ∈ C^{0}(B_{n}(R), t_{n−1}) respectively. Then

g_{i}α_{1}α_{2} = (w_{1}^{−1}g_{i}w_{1})α_{2}
= (w_{1}^{−1}α_{2})(g_{i}α_{2})(w_{1}α_{2})
= (w_{1}^{−1}α_{2})(w_{2}^{−1}g_{i}w_{2})(w_{1}α_{2})

This shows that α_{1}α_{2} is inner at i via w_{2}(w_{1}α_{2}) which is clearly in C^{0}(B_{n}(R), t_{n−1}). The group C^{0}(B_{n}(R), t_{n−1}) is normal in Aut^{0}(F_{n,∞}, t_{n−1}) and so is normal in Aut_{I_{i}}^{0}(F_{n,∞}, t_{n−1}). If α is inner at i via w ∈ C^{0}(B_{n}(R), t_{n−1}), then α′ obtained by composing α with conjugation by w^{−1} is in Aut_{g_{i}}^{0}(F_{n,∞}, t_{n−1}). Any element common to C^{0}(B_{n}(R), t_{n−1}) and Aut_{g_{i}}^{0}(F_{n,∞}, t_{n−1}) is trivial by Proposition 5.3.1. The last provision follows from (b).

(d) We will use superscripts to indicate that certain points are fixed by automorphisms or permutations. The inclusions

⟨Aut_{g_{i}}^{0}(F_{n,∞}, t_{n−1})⟩π_{n} ⊆ ⟨Aut_{I_{i}}^{0}(F_{n,∞}, t_{n−1})⟩π_{n} ⊆ Π_{n}^{0,i,i+1}

are obvious. For the inclusion Π_{n}^{0,i,i+1} ⊆ ⟨Aut_{g_{i}}^{0}(F_{n,∞}, t_{n−1})⟩π_{n} consider σ ∈ Π_{n}^{0,i,i+1} and let τ = ρ_{n−2−i}^{−1}σρ_{n−2−i} ∈ Π_{n}^{0,n−2−i,n−2}. Now τ ∈ Π_{n−1}^{0,n−2−i} and by Theorem 4.4.3 there is an α ∈ Aut^{0}(F_{n−1,∞}, t_{n−2}) such that απ_{n−1} = τ. Since τ fixes (n−2−i), Proposition 1.2.2 gives a W ∈ F_{0,[0,n−2]} taking (n−2−i)α to (n−2−i). If α′ is α composed with conjugation by W Φ_{n−1}, then α′ ∈ Aut^{0,n−2−i}(F_{n−1,∞}, t_{n−2}) and α′π_{n−1} = τ. Let β = α′Θ_{n−1,n}. By Proposition 4.4.2, βπ_{n} = τ, and by Theorem 4.1.6, β ∈ Aut_{g_{n−2}}^{0,n−2}(F_{n,∞}, t_{n−1}). An easy argument using the notion of leftmost break shows that β ∈ Aut_{g_{n−2}}^{0,n−2−i,n−2}(F_{n,∞}, t_{n−1}). By Lemma 4.2.1, the (n − 2 − i)-step rotation of β is γ = t_{n−2−i}^{−1}βt_{n−2−i} so γ is in Aut_{g_{i}}^{0,i,i+1}(F_{n,∞}, t_{n−1}). By Proposition 4.2.2, γπ_{n} = σ.

(e) When n is not a prime power, the second sentence follows from Example 5.3.4 given after Corollary 5.3.3.1. The rest follows from (c) and (d).

From Lemma 5.3.3(a), we get a we get a well-defined injective function

ι_{n} : Aut_{LI}^{0}(F_{n,∞}, t_{n−1}) → F_{n,[0,n−1]}^{n−1}

so that if α ∈ Aut_{LI}^{0}(F_{n,∞}, t_{n−1}) and αι_{n} = (W_{0}, …, W_{n−2}) then for each i with 0 ≤ i ≤ n − 2, we have g_{i}α = w_{i}^{−1}g_{i}w_{i}, where w_{i} = W_{i}Φ_{n}. A further consequence of Lemma 5.3.3(a) is the following.

Corollary 5.3.3.1. An element α ∈ Aut_{LI}^{0}(F_{n,∞}, t_{n−1}) is inner if and only if αι_{n} is in the diagonal of F_{n,[0,n−1]}^{n−1}.

Example 5.3.4. If n is not a prime power, then let p be a prime divisor of n. For some k, q = p^{k} will be congruent to 1 modulo n − 1. By Lemma 3.2.3, there is a PL normalizer h of B_{n}(R) that fixes 0, that commutes with t_{n−1} and that has slopes in q⟨n⟩. The normalizer h is not in B_{n}(R) since q is not in ⟨n⟩. However, it is easy to show that hπ_{n} is trivial. If for some i there were a w ∈ C^{0}(F_{n,∞}, t_{n−1}) so that wh^{−1}g_{n,i}hw^{−1} = g_{n,i}, then the PL normalizer hw^{−1} would be non trivial and commute with g_{n,i} contradicting Proposition 5.3.1. This shows that for our assumed n, the intersection of Aut_{P L}(F_{n,∞}, t_{n−1}) and Aut_{π}^{0}(F_{n,∞}, t_{n−1}) does not lie in Aut_{LI}^{0}(F_{n,∞}, t_{n−1}).

The next lemma shows how lifting affects an automorphism that is inner (at some i). We use a^{b} to represent b^{−1}ab or the action of b on a as appropriate.

Lemma 5.3.5. Let 2 ≤ m < n be integers, and let τ_{m,n} and Θ_{m,n} be as in Theorem 4.1.6. Let W be in F_{m,[0,m−1]}, let W′ = W τ_{m,n}, and let w = W Φ_{m} ∈ C^{0}(B_{m}(R), t_{m−1}) be regarded as an element of Aut^{0}(F_{m,∞}, t_{m−1}).

(a) Then W′ is in F_{n,[0,m−1]} ⊆ F_{n,[0,n−1]} and if u = wΘ_{m,n} ∈ Aut^{0}(F_{n,∞}, t_{n−1}) and v = W′Φ_{n} ∈ C^{0}(B_{n}(R), t_{n−1}) ⊆ Aut^{0}(F_{n,∞}, t_{n−1}), then for all i with 0 ≤ i ≤ m − 2, we have (g_{n,i})^{u} = (g_{n,i})^{v}.

(b) Now let i be an integer with 0 ≤ i ≤ m−2. Assume that α ∈ Aut^{0}(F_{m,∞}, t_{m−1}) is inner at i via w. Let β = αΘ_{m,n}, and let v = W′Φ_{n}. Then (g_{n,i})^{β} = (g_{n,i})^{v} and β is inner at i via v.

Proof. We get (b) from (a) and Corollary 4.1.6.1 because (g_{n,i})^{β} = (g_{n,i})^{u} = (g_{n,i})^{v} with u as in (a). We thus concentrate on (a).

Let W_{1} = W σ_{m−1}. That W′ is in F_{n,[0,m−1]} follows from Lemma 4.1.5. Let

W′_{1} = W′σ_{n−1} = t_{n−1}^{−1}W′t_{n−1} = (t_{m−1}^{−1}W t_{m−1})τ_{m,n} = (W σ_{m−1})τ_{m,n} = W_{1}τ_{m,n}.

Now two applications of Lemma 3.1.2(b) and one of Theorem 4.1.6 give

(g_{n,i})^{v} = (W′)^{−1}g_{n,i}W′W′_{1} = (W^{−1}g_{m,i}W W_{1})τ_{m,n} = ((g_{m,i})^{w})τ_{m,n} = (g_{n,i})^{u}.

Corollary 5.3.5.1. Let m < n, let α be in Aut^{0}(F_{m,∞}, t_{m−1}) and let β = αΘ_{m,n} ∈ Aut^{0}(F_{n,∞}, t_{n−1}). If α is locally inner, then so is β.

Proof. Lemma 5.3.5 works for each i with 0 ≤ i < m − 1. For m − 1 ≤ i < n − 1, let w_{i} = 1.

Now that we know that locally inner automorphisms are closed under lifting, we can define another subgroup of Aut^{0}(F_{n,∞}, t_{n−1}). As n ranges over the integers greater than 1, the groups Aut^{0}(F_{n,∞}, t_{n−1}) form a graded group that we denote by 𝒜. This graded group is closed under lifting. We let ℒ𝒞 be the smallest graded subgroup of 𝒜 that contains all the C^{0}(B_{n}(R), t_{n−1}) and that is closed under lifts. The portion of ℒ𝒞 in Aut^{0}(F_{n,∞}, t_{n−1}) will be denoted LC^{0}(B_{n}(R), t_{n−1}). The next corollary follows from Corollary 5.3.5.1.

Corollary 5.3.5.2. LC^{0}(B_{n}(R), t_{n−1}) is a subgroup of Aut_{LI}^{0}(F_{n,∞}, t_{n−1}).

5.4. Normalizers of A_{n}. In this section, we obtain our main results from the machinery developed in Section 5.3. The results here will prove that certain normalizers of A_{n} are not PL and that certain normalizers of B_{n} are not normalzers of A_{n}.

Let W ∈ F_{n,∞} and 0 ≤ j < n − 1. We say that W avoids (the residue class of) j if W can be written as a product of generators in Definition 2.1.1 and their inverses that uses no g_{i} with i congruent to j modulo n − 1. Let w ∈ C^{0}(B_{n}(R), t_{n−1}). We say that w avoids (the residue class of) j if we can write w = W Φ_{n} for some W ∈ F_{n,[0,n−1]} such that W avoids j.

Note that if W avoids j then it can be written in seminormal form without using generators whose subscripts are congruent to j (mod n − 1). This is due to the fact that the rewriting rules that lead to the seminormal form preserve the residue classes of the subscripts.

Lemma 5.4.1. (a) {W ∈ F_{n,∞}|W avoids j} is a subgroup of F_{n,∞}.

(b) {w ∈ C^{0}(B_{n}(R), t_{n−1})|w avoids j} is a subgroup of C^{0}(B_{n}(R), t_{n−1}).

(c) Let m < n, W ∈ F_{m,∞} and τ_{m,n} : (F_{m,∞}, t_{m−1}) → (F_{n,∞}, t_{n−1}) be the monomorphism of Proposition 4.1.2. Then W τ_{m,n} avoids every j with m − 1 ≤ j < n − 1.

Proof. (a) It is immediate from the definition.

(b) This follows from (a) and Lemma 3.1.2(a).

(c) This is immediate from the definition of τ_{m,n} in Proposition 4.1.2.

Lemma 5.4.2. Let W ∈ F_{n,∞}, and 0 ≤ j < n − 1. Then W avoids j if and only if W can be represented by two allowable subdivisions in the sense of Lemma 2.2.2 such that for every l ≥ 0, the intervals [\frac{a}{n^{l}} , \frac{a+1}{n^{l}}], a ≡ j (mod n − 1), are not subdivided.

Proof. Note that when an allowable subdivision is constructetd as in Definition 2.2.3, the number of an interval of the form [\frac{a}{n^{l}} , \frac{a+1}{n^{l}}] is congruent to a (mod n−1), so each generator g_{i} in the seminormal form of W will partition an interval [\frac{a}{n^{l}} , \frac{a+1}{n^{l}}] with a ≡ i (mod n − 1).

Corollary 5.4.2.1. Let a < b < c be in Z, 0 ≤ j < n − 1, and W_{1}, W_{2} ∈ F_{n,∞} with supp(W_{1}) ⊆ [a, b], supp(W_{2}) ⊆ [b, c]. If W_{1}W_{2} avoids j, then so do W_{1} and W_{2}.

Proof. Write W_{1}W_{2} = P N^{−1} in seminormal form so that the allowable subdivisions D and E corresponding to P and N respectively, do not subdivide intervals of the form [\frac{a}{n^{l}} , \frac{a+1}{n^{l}}] with a ≡ j (mod n − 1). Combining the portions of D and E that lie in (−∞, b] with the standard subdivision on [b, ∞) yields a pair of allowable subdivisions for W_{1}. Similarly, the portions of D and E in [b, ∞) combined with the standard subdivision on (−∞, b] yield a pair of allowable subdivisions for W_{2}. Therefore W_{1} and W_{2} avoid j.

Lemma 5.4.3. Let b ∈ Z, 0 ≤ j < n − 1, and w ∈ C^{0}(B_{n}(R), t_{n−1}) be such that w fixes b. Then t_{b}wt_{b}^{−1} ∈ C^{0}(B_{n}(R), t_{n−1}) and w avoids j iff t_{b}wt_{b}^{−1} avoids j − b.

Proof. Since w commutes with t_{n−1} we may assume that 0 ≤ b < n − 1. The fact that t_{b}wt_{b}^{−1} ∈ C^{0}(B_{n}(R), t_{n−1}) is immediate. For the rest, it suffices to do one direction. Assume w avoids j. Write w = W Φ_{n} with W ∈ F_{n,[0,n−1]}, so that W avoids j. Then we have that W fixes b. So write W = W_{1}W_{2} where supp(W_{1}) ⊆ [0, b] and supp(W_{2}) ⊆ [b, n − 1]. By Corollary 5.4.2.1 both W_{1} and W_{2} avoid j. Let U_{1} = t_{b}W_{2}t_{b}^{−1} and U_{2} = t_{n−1−b}^{−1}W_{1}t_{n−1−b}, and U = U_{1}U_{2}. Clearly both U_{1} and U_{2} avoid j − b so by Lemma 5.4.1(a), U avoids j − b. Since we have t_{b}wt_{b}^{−1} = UΦ_{n} the conclusion is that t_{b}wt_{b}^{−1} avoids j − b.

Lemma 5.4.4. For n ≥ 3, let α be in Aut^{0}(F_{n,∞}, t_{n−1}), and let h be the selfhomeomorphism of R that fixes 0, commutes with t_{n−1} and realizes α. Let 0 ≤ i < n − 1 and γ_{i}, γ_{i+1} ∈ Aut_{P L}(B_{n}(R), t_{n−1}) be conjugation by PL homeomorphisms w_{i} and w_{i+1} such that (g_{i})α = (g_{i})γ_{i} and (g_{i+1})α = (g_{i+1})γ_{i+1}. If h normalizes A_{n}(R), then h^{−1}t_{1}h = w_{i}^{−1}t_{1}w_{i+1}.

Proof. Observe that (g_{i})γ_{i}α^{−1}σ_{1}αγ_{i+1}^{−1}σ_{1}^{−1} = g_{i}. Since h normalizes A_{n}(R), both h^{−1}t_{1}h and w_{i}h^{−1}t_{1}hw_{i+1}^{−1}t_{1}^{−1} are PL. Applying Proposition 5.3.1 to this PL normalizer of B_{n}(R) which commutes with t_{n−1} we get w_{i}h^{−1}t_{1}hw_{i+1}^{−1}t_{1}^{−1} = 1, so h^{−1}t_{1}h = w_{i}^{−1}t_{1}w_{i+1}.

Proposition 5.4.5 (Double unbent generator). Let α ∈ Aut^{0}(F_{n,∞}, t_{n−1}), n ≥ 3, be a normalizer of A_{n}(R). If for some 0 ≤ i < n − 1, α has an unbent generator at i and i + 1, then α is the identity.

Proof. Let h be the self-homeomorphism of R that fixes 0, commutes with t_{n−1} and realizes α. By Corollary 5.2.2.1, h^{−1}g_{i}h has slope n to the right of ih, and being affine on [i, i+1]h, forces this interval to have width 1. Similarly for [i+1, i+2]h, and the two intervals are consecutive. Let b = ih−i = (i+1)h−(i+1) = (i+2)h−(i+2). Then h^{−1}g_{i}h = t_{b}^{−1}g_{i}t_{b} and h^{−1}g_{i+1}h = t_{b}^{−1}g_{i+1}t_{b}. By Lemma 5.4.4, h^{−1}t_{1}h = t_{1}, so h commutes with t_{1}. It follows that h^{−1}g_{j}h = t_{b}^{−1}g_{j}t_{b} for every 0 ≤ j < n − 1, so α is conjugation by t_{b}. Since h fixes 0, we must have b = 0 so α ≡ 1 .

Lemma 5.4.6. For n ≥ 3, let α ∈ Aut^{0}(F_{n−1,∞}, t_{n−2}) be locally inner. If β = αΘ_{n−1,n} normalizes A_{n}(R) then α is the identity.

Proof. By Lemmas 5.4.1(c) and 5.3.5, for each i there are w_{i} ∈ C^{0}(B_{n}(R), t_{n−1}) such that g_{i}β = w_{i}^{−1}g_{i}w_{i} and w_{i} avoids n−2. Moreover w_{n−2} = 1 and w_{i+n−1} = w_{i}. It follows from Lemma 5.4.4 that h^{−1}t_{1}h = w_{i}^{−1}t_{1}w_{i+1}, so we get h^{−1}t_{1}h = t_{1}w_{0} = w_{0}^{−1}t_{1}w_{1} = ⋯ = w_{n−4}^{−1}t_{1}w_{n−3} = w_{n−3}^{−1}t_{1} so

(∗)  w_{i+1} = t_{1}^{−1}w_{i}t_{1}w_{0} for 0 ≤ i ≤ n − 4,
(∗∗)  1 = t_{1}^{−1}w_{n−3}t_{1}w_{0}.

From (∗) and the fact that w_{0} and w_{i+1} fix 0, we get that w_{i} fixes n − 2 when 0 ≤ i ≤ n − 4. For w_{n−3}, use (∗∗) and we have that all w_{i} fix n − 2.

Now from (∗∗), Lemmas 5.4.1(b) and 5.4.3, and the fact that w_{n−3} avoids n − 2, we get that w_{0} avoids 0. From this, (∗), Lemmas 5.4.1(b) and 5.4.3, and the fact that all w_{i} avoid n − 2, we get that all w_{i} avoid 0. Once again, iterate the process to get that all w_{i} avoid each of 0, 1, …, n − 2. Therefore we must have all w_{i} = 1 and α ≡ 1.

Theorem 5.4.7. For 2 ≤ m < n, let α ∈ Aut^{0}(F_{m,∞}, t_{m−1}). For m = n − 1 assume that α is locally inner. If β = αΘ_{m,n} normalizes A_{n}(R) then α is the identity.

Proof. This follows from Lemma 5.4.6, when m = n − 1, and from Proposition 5.4.5, when m < n − 1.

The next application of Proposition 5.4.5 justifies the remark made after Theorem 4.3.1.

Theorem 5.4.8. Assume (m−1) | (n−1) with m > 2. Let w ∈ C^{0}(B_{m}(R), t_{m−1}) represent a non-trivial automorphism α of A_{m}(R) by conjugation. Let β = wΛ_{m,n} as described in Theorem 4.3.1. Then β does not normalize A_{n}(R).

Proof. Assume that β normalizes A_{n}(R). We have w = W Φ_{m} for some W ∈ F_{m,[0,m−1]}. Now if W′ = W τ_{m,n}, then by Lemma 4.1.5, we have W′ ∈ F_{n,[0,m−1]} ⊆ F_{n,[0,n−1]}. If w′ = W′Φ_{n}, then Lemma 5.3.5 (which applies since β and αΘ_{m,n} agree on the relevant generators) gives g_{n,i}β = (w′)^{−1}g_{n,i}w′ for 0 ≤ i ≤ (m − 2). Now w′ avoids j unless 0 ≤ jϕ_{n} ≤ (m − 2). Thus if w′ were to commute with t_{m−1}, then w′ would be trivial. Since β fixes t_{m−1}, the self homeomorphism h of R for which g_{n,j}β = h^{−1}g_{n,j}h for all j must commute with t_{m−1} as well. Thus h ≠ w′. Now the composition (w′)^{−1}h is non-trivial and normalizes A_{n}(R) since w′ ∈ A_{n}(R). However, conjugation by (w′)^{−1}h leaves each g_{n,i} fixed with 0 ≤ i ≤ (m − 2) which contradicts Proposition 5.4.5.

The next theorem will be used to show that there are many non-PL elements in Aut(A_{n}(R)).

Theorem 5.4.9. Let 2 ≤ m < n be integers for which (m − 1) | (n − 1), and consider α ∈ Aut_{LI}^{0}(F_{m,∞}, t_{m−1}). Let β = αΛ_{m,n} as described in Theorem 4.3.1. If α is not trivial, then β is not PL.

Proof. From the construction in Theorem 4.3.1, we know that β = αΘ_{m,n} on those g_{n,i} with 0 ≤ i ≤ m − 2. By Lemma 5.3.5, β is inner at i via some w_{i} in C^{0}(B_{n}(R), t_{n−1}) for all i with 0 ≤ i ≤ m − 2. For any j, g_{n,j} = t_{m−1}^{−k}g_{n,i}t_{m−1}^{k} for some unique k and i with 0 ≤ i ≤ m − 2. An elementary calculation using the fact that β commutes with t_{m−1} shows that β is inner at j via w_{j} where w_{j} is equal to the conjugate of w_{i} by t_{m−1}^{k}.

By Lemma 5.3.3(e), we are done if we show that β is not inner. By Corollary 5.3.3.1, we are done if we show that β is inner at two values via unequal elements of C^{0}(B_{n}(R), t_{n−1}).

Consider an i with 0 ≤ i ≤ m−2. Since β = αΘ_{m.n} on g_{n,i}, Lemma 5.4.1(c) says that w_{i} avoids all j with m − 1 ≤ j < n − 1. Now β is inner at i + m − 1 by w′_{i} = t_{m−1}^{−1}w_{i}t_{m−1}. By Lemma 5.4.3, w′_{i} avoids all j with 2(m−1) ≤ j < (n−1)+(m−1) modulo (n − 1). However, this includes all j with 0 ≤ j < m − 1 modulo (n − 1). If we were to have w_{i} = w′_{i}, we would have w_{i} = 1. This cannot happen for all i with 0 ≤ i ≤ m − 2 since α is not trivial. This competes the proof.

5.5. An example of torsion. In the following, we leave to the reader the task of checking a large number of calculations.

Theorem 5.5.1. For n ≥ 5 there is an element of order n − 2 in Out_{o}(B_{n}(R)).

Proof. The example will be a rotation of a lift of an element in Aut^{0}(F_{4,∞}, t_{3}). Consider the function fixing t_{3} and sending

\begin{matrix}g_{4,0} & ↦ g_{4,0}g_{4,4}g_{4,2}^{−1}, \\ g_{4,1} & ↦ g_{4,2}g_{4,3}g_{4,5}^{−1}, \\ g_{4,2} & ↦ g_{4,2}g_{4,4}g_{4,2}^{−1}. \end{matrix}

This extends to a function defined on all the g_{4,i} that fixes t_{3}. Using the subdivisions of Section 2.2, it is easy to show that the the images of the g_{4,i}, i ∈ {0, 1, 2}, have their active parts confined to [0, 2], [2, 2\frac{1}{2}] and [2\frac{1}{2}, 3] in order, and that they satisfy the relations defining F_{4,∞}. Thus the function extends to an endomorphism α of F_{4,∞} that commutes with σ_{3}. It is straightforward to check that the inverse of α is defined by

\begin{matrix}g_{4,0} & ↦ g_{4,0}g_{4,1}g_{4,3}^{−1}, \\ g_{4,1} & ↦ g_{4,0}g_{4,2}g_{4,0}^{−1}, \\ g_{4,2} & ↦ g_{4,1}g_{4,5}g_{4,3}^{−1}. \end{matrix}

It is not needed here, but α is the automorphism used in Theorem 4.4.3 that is conjugation by a PL function with slopes in 2⟨4⟩ and thus α is not in C^{0}(F_{4,∞}, t_{3}).

If we lift α to β_{n} by Θ_{4,n} for n > 4, then we get an automorphism defined by

\begin{matrix}g_{n,0} & ↦ g_{n,0}g_{n,n}g_{n,2}^{−1}, \\ g_{n,1} & ↦ g_{n,2}g_{n,n−1}g_{n,n+1}^{−1}, \\ g_{n,2} & ↦ g_{n,2}g_{n,n}g_{n,2}^{−1}, \\ g_{n,i} & ↦ g_{n,i} for 3 ≤ i ≤ n − 2. \end{matrix}

If h is the self homeomorphism of R that realizes β_{n} by conjugation, then the 1-step rotation γ_{n} of β_{n} is realized as conjugation by k = t_{1}ht_{1h}^{−1}. However, 1h = 2 since it can be checked that the leftmost break of g_{n,1}β_{n} is 2. Thus γ_{n} is conjugation by t_{1}ht_{2}^{−1} and γ_{n} = σ_{1}β_{n}σ_{2}^{−1}. This gives that γ_{n} is determined by

\begin{matrix}g_{n,0} ↦ & g_{n,0}g_{n,n−3}g_{n,n−1}^{−1}, \\ g_{n,1} ↦ & g_{n,0}g_{n,n−2}g_{n,0}^{−1}, \\ g_{n,i} ↦ & g_{n,i−1} for 2 ≤ i ≤ n − 3, \\ g_{n,n−2} ↦ & g_{n,n−3}g_{n,2n−3}g_{n,n−1}^{−1} \\ = & g_{n,n−2}g_{n,n−3}g_{n,n−1}^{−1}. \end{matrix}

where the last expression that is not in seminormal form will sometimes be convenient.

Let

\begin{aligned}u_{n,k} & = g_{n,0}g_{n,n−k}g_{n,n−1}^{−1}, and \\ v_{n,k} & = g_{n,n−2}g_{n,n−k}g_{n,n−1}^{−1}. \end{aligned}

Note that g_{n,0}γ_{n} = u_{n,3} and g_{n,n−2}γ_{n} = v_{n,3}. We claim that u_{n,k}γ_{n} = u_{n,k+1} and v_{n,k}γ_{n} = v_{n,k+1} for 3 ≤ k ≤ n − 2. In this range, 2 ≤ n − k ≤ n − 3 and

u_{n,k}γ_{n} = g_{n,0}g_{n,n−3}g_{n,n−1}^{−1}g_{n,n−(k+1)}g_{n,2n−2}g_{n,2n−4}^{−1}g_{n,n−1}^{−1}
= g_{n,0}g_{n,n−(k+1)}g_{n,n−1}^{−1}
= u_{n,k+1}

where the second equality holds because n − (k + 1) is less than both n − 1 and n − 3. The claim for v_{n,k}γ_{n} has an almost identical proof.

Now g_{n,n−3}γ_{n}^{j} = g_{n−3−j} for 1 ≤ j ≤ n − 4. Thus, for these values of j, γ_{n}^{j} is non-trivial and satisfies the hypotheses of Proposition 5.3.1. For these values of j we know that γ_{n}^{j} is not represented as conjugation by a PL self homeomorphism of R and represents a non-trivial element in Out_{o}(B_{n}(R)). When we show that γ_{n}^{n−2} represents a trivial element in Out_{o}(B_{n}(R)), we will know that γ_{n}^{n−3} represents a non-trivial element in Out_{o}(B_{n}(R)) as well.

Now

g_{n,0}γ_{n}^{n−2} = u_{n,n−1}γ_{n}
= (g_{n,0}g_{n,1}g_{n,n−1}^{−1})γ_{n}
= g_{n,0}^{2}g_{n,n−2}g_{n,2n−2}^{−1}g_{n,0}^{−1}

where the last equality follows from the definition of γ_{n} and several applications of the relations of F_{n,∞}. A similar calculation gives

g_{n,n−2}γ_{n}^{n−2} = v_{n,n−1}γ_{n} = g_{n,0}g_{n,n−2}g_{n,3n−4}g_{n,2n−2}^{−1}g_{n,0}^{−1}.

To help work with the other generators, we let P = g_{n,n−2}g_{n,0}^{−1}. An easy check shows P γ_{n} = P. Now g_{n,1}γ_{n} = g_{n,0}P so for 1 ≤ i ≤ n − 3 we have

g_{n,i}γ_{n}^{n−2} = (g_{n,0}P)γ_{n}^{n−2−i} = u_{n,n−i}P
= g_{n,0}g_{n,i}g_{n,n−1}^{−1}g_{n,n−2}g_{n,0}^{−1}
= g_{n,0}g_{n,i}g_{n,n−2}g_{n,2n−2}^{−1}g_{n,0}^{−1}.

We will show that γ_{n}^{n−2} is trivial in Out_{o}(B_{n}(R)) by showing that the action of γ_{n}^{n−2} is realized as conjugation by an element of C^{0}(F_{n,∞}, t_{n−1}). We claim that the action of γ_{n}^{n−2} is conjugation by w = P Φ_{n}. Using subdivisions, it is easy to check that P has support in [0, n − 1] so that P Φ_{n} makes sense. By Lemma 3.1.2(b), w^{−1}g_{n,i}w = P^{−1}g_{n,i}P P_{1} where P_{1} = P σ_{n−1} = g_{n,2n−3}g_{n,n−1}^{−1}. It is now a straightforward exercise in using the relations of F_{n,∞} to check that for 0 ≤ i ≤ n−2 we get w^{−1}g_{n,i}w = g_{n,i}γ_{n}^{n−2}. This completes the proof.

The lemma above holds for n = 4 as well. The example is represented by the automorphism α in the proof. To verify this, one must actually prove the remark in the proof that the action of α is conjugation by a PL function with slopes in 2⟨4⟩. The order of this outer automorphism is 2. What is surprising is that the finiteness of the order persists in the particular combination of lift and rotation used. In fact, all j-step rotations of β_{n}, n > 4, by integers with ϕ_{n}(j) ≠ 1 and ϕ_{n}(j) ≠ 2 represent outer automorphisms of infinite order. This follows from Proposition 5.3.1 since each such rotation fixes g_{n,0} and is non-trivial (and thus its positive powers are non-trivial). We do not know the status of the 2-step rotation of β_{n}, although numerical calculations hint that the order is infinite.

6. On the structure of Aut(A_{n}) and Aut(B_{n})

In this section we collect and comment on the main results of the paper. Section 6.1 gives general results about the automorphisms of A_{n} and B_{n} for a single value of n. Section 6.2 discusses local properties of automorphisms, i.e., the action on a single generator. Section 6.3 discusses the relations between automorphims of A_{m} and A_{n}, and B_{m} and B_{n} for values 2 ≤ m < n. The results are not arranged to show dependency. All results can be proven using material prior to this section, but in the order that they are presented here, some results in Section 6.1 depend on results quoted in Sections 6.2 and 6.3.

6.1. Results for a single n. The classes [A_{n}] and [B_{n}], n ≥ 2, of homeomorphism groups on both the real line, and the circle are defined in Section 1.3. For C = A or C = B, the class [C_{n}] has a container group C_{n}. The McCleary-Rubin theorem applies to all groups in the classes, automorphisms can be viewed as normalizers, and the characteristic subgroups of C_{n} are precisely the ones with the same automorphism group as C_{n}.

The next two lemmas are minor rephrasing of Lemmas 1.3.1, 1.3.2 and 2.2.7.

Lemma 6.1.1. Let X = R or X = S_{r}, C = A or C = B, and r ∈ Z[\frac{1}{n}] with r ∈ Δ_{n} for the case C = B. Then the following hold.

(a) The groups C_{n}(R), C_{n}(S_{r}), are in the class [C_{n}].

(b) For G in the class [C_{n}], N(G) ≃ Aut(G).

(c) If G is in [C_{n}], we have N(G) ⊆ N(C_{n}(X)).

(d) G ∈ [C_{n}] is characteristic in C_{n}(X) if and only if N(G) = N(C_{n}(X)).

Each of the two classes contains at least one of the Thompson groups.

Lemma 6.1.2. The group F_{n} is in [A_{n}], and the group F_{n,∞}, is in [B_{n}].

Moreover, in each case groups of automorphisms of the container and the corresponding Thompson group are closely related.

Proposition 6.1.3. There are exact sequences and isomorphisms

(1) 1 → Aut_{B}(F_{n,∞}) → Aut_{o}(F_{n,∞}) → Out_{o}(B_{n}(R)) → 1,

(2) 1 → Aut_{B}^{0}(F_{n}) → Aut_{π}^{0}(F_{n}) → Out_{π}(A_{n}(R)) → 1,

(3) Out_{π}(B_{n}(R)) ≃ Aut_{π}^{0}(F_{n,∞}, t_{n−1})/C^{0}(B_{n}(R), t_{n−1}),

(4) Out_{π}(A_{n}(R)) ≃ Aut_{π}^{0}(F_{n}, t_{1})/C^{0}(B_{n}(R), t_{1}),

\xymatrix{1 \ar[r] & F_{n,∞}^{0} \ar[r] & Aut_{B}^{0}(F_{n}) \ar[r]^{θ_{A}} & T_{n,1} × T_{n,1} \ar[r] & 1}, and (5)

\xymatrix{1 \ar[r] & F_{n,∞} \ar[r] & Aut_{B}(F_{n,∞}) \ar[r]^{θ_{B}} & T_{n,n−1} × T_{n,n−1} \ar[r]^{d} & Z_{n−1} \ar[r] & 1.} (6)

Note that the middle terms of (5) and (6) are the second terms of (2) and (1).

Proof. Items (1–4) are given in Proposition 2.6.2. In (5), the restriction that elements of F_{n} must be integer translations near ±∞ forces an element f of Aut_{B}^{0}(F_{n}) to satisfy (x + 1)f = xf + 1 near ±∞. We thus take fθ_{A} to be the pair of germs of f near ±∞ modulo integer translations. This gives a well defined pair of elements of T_{n,1}. If such a pair of elements is given, then they lift to a pair (f_{−}, f_{+}) where each is in Aut(F_{n}). Since the lifts are adjustable by composing with translations by integers, we can arrange that f_{−}ρ_{n} = f_{+}ρ_{n} = 0 and that xf_{+} > x and xf_{−} < x for all x. Then Proposition 1.2.2 lets us build an element f of A_{n}(R) that agrees with f_{±} near ±∞ and that fixes 0. This shows that θ_{A} is onto. The kernel of θ_{A} consists of all elements of B_{n}(R) that fix 0 and are integer translations near ±∞. Since residues are preserved in B_{n}(R), the translations are by multiples of of n − 1 and the kernel consists of all elements of F_{n,∞} that fix 0.

The arguments for (6) are similar to those for (5) except that we cannot adjust f_{−}ρ_{n} and f_{+}ρ_{n}. In fact there is a well defined homomorphism ρ_{n} : T_{n,n−1} → Z_{n−1} giving the amount of shift on residue classes induced by an element of T_{n,n−1}. The image of θ_{B} is the kernel of the map d : T_{n,n−1} × T_{n,n−1} → Z_{n−1} defined by (f_{1}, f_{2})d = f_{1}ρ_{n} − f_{2}ρ_{n}.

The two groups A_{n}(R) and B_{n}(R) have the same PL normalizers which are completely determined by Lemma 3.2.2. From Lemma 3.2.2 and Corollary 3.2.4.1 we have:

Theorem 6.1.4. Let n ≥ 2.

(a) Aut_{P L}(A_{n}(R)) = Aut_{P L}(B_{n}(R))

Out_{P L}(A_{n}(R)) ≃ ⟨⟨n⟩⟩/⟨n⟩ (b)

Out_{P L}(B_{n}(R)) ≃ Z_{n−1} ⋊ ⟨⟨n⟩⟩/⟨n⟩. (c)

See Corollary 3.2.4.2 for the analysis of torsion that occurs in Out_{P L}(A_{n}(R)) and Out_{P L}(B_{n}(R)).

From Theorem 5.4.9, we get infinitely many non-PL automorphisms of A_{n}.

Theorem 6.1.5. For 3 ≤ n, Aut(A_{n}(R))contains an isomorphic copy F̂ of F ≃ F_{2} ≃ F_{2,∞}. The intersection of F̂ with Aut_{P L}(A_{n}(R)) is trivial.

Proof. Theorem 4.3.1 says that Λ_{2,n} injects Aut^{0}(F_{2,∞}, t_{1}) into Aut^{0}(F_{n,∞}, t_{1}). Since F_{n,∞} and t_{1} generate F_{n}, Λ_{2,n} injects Aut^{0}(F_{2,∞}, t_{1}) into Aut^{0}(F_{n}, t_{1}). Since automorphisms of F_{n} are automorphisms of A_{n}(R), Λ_{2,n} injects Aut^{0}(F_{2,∞}, t_{1}) into Aut^{0}(A_{n}(R), t_{1}). From Theorem 5.4.9, we get that the image of the nontrivial locally inner elements in Aut^{0}(F_{2,∞}, t_{1}) are not PL. However, elements of Aut^{0}(F_{2,∞}, t_{1}) are orientation preserving, and it follows from the main theorem of [2] that all elements of Aut^{0}(F_{2,∞}, t_{1}) are in C^{0}(B_{2}(R), t_{1}) and are thus (locally) inner. Further C^{0}(B_{2}(R), t_{1}) is isomorphic to F_{0,[0,1]} which is isomorphic to each of F_{2,∞}, F_{2} and Thompson’s original group F.

The two container groups and their automorphism groups present some unusual behavior in the sense that A_{n}(R) and B_{n}(R) “almost equal” whereas Aut(A_{n}(R)) and Aut(B_{n}(R)) are “very different”. More precisely:

Theorem 6.1.6. Let n > 2, A = A_{n}(R), B = B_{n}(R). The following properties hold:

(a) B is characteristic in A so that restriction to B gives a homomorphism from Aut(A) to Aut(B).

(b) The homomorphism from Aut(A) to Aut(B) is a monomorphism so that we can regard Aut(A) as a subgroup of Aut(B).

(c) The index of B in A is finite.

(d) The index of Aut(A) in Aut(B) is infinite.

Proof. The first three parts follow from Proposition 1.4.2 and Lemma 1.1.4. For part (d), Theorem 5.4.7 implies that Aut(A) has trivial intersection with the infinite groups ⟨Aut^{0}(F_{m,∞}, t_{m−1})⟩Θ_{m,n} for m ≤ n − 2 and ⟨Aut_{LI}^{0}(F_{n−1,∞}, t_{n−2})⟩Θ_{n−1,n}.

The behavior in Theorem 6.1.6 although unusual, is found in other places. The following example is a variation of one supplied to us by F. Gross [15]. Note that while there are other groups having the properties imposed on S in the hypotheses, the group F_{n}^{s} of Proposition 2.4.2 will also do when n > 2.

Proposition 6.1.7. Let S be a simple group such that Out(S) is infinite, and P a finite group acting transitively on the set {1, …, n}, n ≥ 2. Let B = S^{n} and A = S ≀P, so that A = B ⋊P. Then A and B satisfy the four properties of Theorem 6.1.6

Proof. (a) This follows from from [21, Thm 9.2] since S is not a dihedral group. However it can also be seen directly using the facts that S is infinite simple and P is finite as follows: Let α ∈ Aut(A), and let S_{i} denote the i-th factor of B. Since by construction B is a normal subgroup of A we have that Bα ∩ S_{i} is a normal subgroup of S_{i}, so by simplicity we have Bα ∩ S_{i} = S_{i} or 1. But [S_{i} : Bα ∩ S_{i}] = [(Bα)S_{i} : Bα] ≤ [A : Bα] = n so we must have Bα ∩ S_{i} = S_{i}, i.e., S_{i} ⊆ Bα. Therefore B ⊆ Bα. Now use α^{−1} to get equality.

(c) Is obvious. In the rest of the proof, it will be convient at times to use a^{b} to denote b^{−1}ab when a and b are in the same group, and the action of b on a when a ∈ G and b ∈ Aut(G). We let : Aut(A) → Aut(B) be the monomorphism induced by restriction. By [23, Thm. 3.3.20] we know that Aut(B) is the wreath product Aut(S) ≀ Π_{n} = (Aut(S))^{n} ⋊ Π_{n} where Π_{n} is acting on the set {1, …, n}.

Observe that for b ∈ B and p ∈ P, we have bp = pb^{p}, so for α ∈ Aut(A) if we write p^{α} = aq where a ∈ B and q ∈ P, then applying α on both sides of bp = pb^{p} we get b^{α}aq = aqb^{pα} = ab^{pαq^{−1}}q, so b^{α}a = ab^{pαq^{−1}}, and therefore

α̂ξ_{a} = pα̂q^{−1} (∗)

where ξ_{a} is conjugation by a in B.

(b) Let α ∈ Aut(A) be such that α̂ = 1. From (∗) we get ξ_{a} = pq^{−1}, and since ξ_{a} ∈ (Aut(S))^{n} and pq^{−1} ∈ Π_{n}, we must have pq^{−1} = 1 and ξ_{a} = 1 so q = p, and a is in the center of B, which is trivial, so p^{α} = p, and α = 1.

(d) Let Δ_{Aut(S)} be the diagonal in (Aut(S))^{n}. We will show that when viewed as a subgroup of Aut(B) we have Aut(A) ⊆ (Δ_{Aut(S)} · (Inn(S))^{n}) ⋊ Π_{n}, and therefore [Aut(B) : Aut(A)] ≥ |Out(S)|^{n−1}. For α ∈ Aut(A), if we write α̂ = δσ where δ = (δ_{1}, …, δ_{n}) ∈ (Aut(S))^{n} and σ ∈ Π_{n}, then from (∗) we get δσξ_{a} = pδσq^{−1} so δξ_{a}^{σ^{−1}}σ = δ^{p^{−1}}pσq^{−1}, and since δξ_{a}^{σ^{−1}}, δ^{p^{−1}} ∈ (Aut(S))^{n} and σ, pσq^{−1} ∈ Π_{n} we get δξ_{a}^{σ^{−1}} = δ^{p^{−1}} so δ^{−1}δ^{p^{−1}} ∈ (Inn(S))^{n}. Finally, since the action of P on {1, …, n} is transitive, δ_{1}, δ_{2}, …, δ_{n} are all in the same coset of Out(S), and we can write (δ_{1}, δ_{2}, …, δ_{n}) = (δ_{1}, δ_{1}, …, δ_{1}) · (1, δ_{1}^{−1}δ_{2}, …, δ_{1}^{−1}δ_{n}) ∈ Δ_{Aut(S)} · (Inn(S))^{n}.

The “difference” between Aut(A_{n}(R)) and Aut(B_{n}(R)) is further illustrated by looking at the permutations in Π_{n−1} that they support. Recall that Aff(Z_{n−1}) denotes the group of affine transformations of Z_{n−1}, i.e., Z_{n−1} ⋊ U_{n−1}, and D_{n−1} is the subgroup of U_{n−1} generated by the divisors of n.

Theorem 6.1.8. Let n > 2.

(a) The homomorphism π_{n} : Aut(B_{n}) → Π_{n−1} is surjective.

(b) The image of Aut_{P L}(A_{n}) under π_{n} equals Z_{n−1} ⋊ D_{n−1}.

(c) The image of Aut(A_{n}) under π_{n} contains Z_{n−1} ⋊ D_{n−1} and is contained in Aff(Z_{n−1}).

Proof. (a) follows immediately from Theorem 4.4.3 and Lemmas 6.1.1 and 6.1.2.

(b) For one containment, Lemmas 3.2.1 and 3.2.2 imply that ⟨Aut_{P L}(A_{n})⟩π_{n} ⊆ Z_{n−1} ⋊ D_{n−1}. Lemmas 3.2.2 and 3.2.3 give the other inclusion.

(c) For the first containment, Z_{n−1} ⋊ D_{n−1} = ⟨Aut_{P L}(A_{n})⟩π_{n} ⊆ ⟨Aut(A_{n})⟩π_{n}. Now for the second containment, given α ∈ Aut(A_{n}(R)), let a = 0α, and k be the self homeomorphism of R that realizes α. Let h = kt_{a}^{−1} and let β be conjugation by h. Then β ∈ Aut^{0}(A_{n}), and since t_{a} ∈ A_{n}(R), then by Lemma 1.1.3, t_{a}π_{n} is a power of the cyclic permutation σ = (0 1 2 … n − 2), which is in Aff(Z_{n−1}). So it suffices to show that γ = βπ_{n} ∈ Aff(Z_{n−1}). Let T = (t_{1})β. On one hand, since T ∈ A_{n}(R) we have T π_{n} is a power of σ. On the other hand, since T = h^{−1}t_{1}h we have that T π_{n} is γ^{−1}σγ an (n − 1)-cycle, so T π_{n} = σ^{j} for some j relatively prime to (n − 1). So T π_{n} = (0 j 2j … (n − 2)j), and since 0γ = 0, we must have

γ = (\begin{matrix}0 & 1 & 2 & … & n − 2 \\ 0 & j & 2j & … & (n − 2)j \end{matrix})

which is multiplication by j modulo n − 1, so γ ∈ Aff(Z_{n−1}).

We end this part by quoting two results about Aut(B_{n}(R)) and Out(B_{n}(R)). First, Aut(B_{n}(R)) is complete (has trivial center and trivial outer automorphism group so that Aut^{2}(B_{n}(R)) = Aut(B_{n}(R))), and second, Out(B_{n}(R)) has torsion. These are Theorems 2.4.3 and 5.5.1 together with the comment following the proof of Theorem 5.5.1.

[figure: Aut^{0}(); 1; Aut_{π}^{0}(); 2; Aut_{I_{i}}^{0}(); Aut_{LI}^{0}(); 3; Aut_{g_{i}}^{0}(); LC^{0}(); 4; C^{0}(); 1]

Figure 2. Subgroups of the automorphism groups

Theorem 6.1.9. The automorphism group of Aut(B_{n}(R)) equals Aut(B_{n}(R)).

Theorem 6.1.10. For n ≥ 4 there is an element of order n − 2 in Out(B_{n}(R)).

6.2. Local properties of automorphisms. Most of the analysis in Section 5 comes from studying the behavior of automorphisms on single generators. This leads to the concepts unbent generator at i, inner, inner at i and locally inner in Definition 5.3.2.

The next two propositions tell us that, other than the identity, PL normalizers of F_{n,∞} can not have unbent generators, and normalizers of F_{n,∞} with two consecutive unbent generators can not normalize A_{n}. These results are Propositions 5.3.1 and 5.4.5.

Proposition 6.2.1 (Unbent Generator). For n ≥ 2, let α ∈ Aut^{0}(F_{n,∞}, t_{n−1}) have an unbent generator at some i. If α is PL, then α is the identity.

Proposition 6.2.2 (Double unbent generator). Let α ∈ Aut^{0}(F_{n,∞}, t_{n−1}), n ≥ 3, be a normalizer of A_{n}(R). If for some 0 ≤ i < n − 1, α has an unbent generator at i and i + 1, then α is the identity.

Section 5.3 introduces several subgroups of Aut^{0}(F_{n,∞}, t_{n−1}). In the subgroup diagram in Figure 2, we use empty parentheses to denote (F_{n,∞}, t_{n−1}) after “Aut” and to denote (B_{n}(R), t_{n−1}) after “C^{0}.”

The next proposition summarizes what we know about the diagram in Figure 2.

Proposition 6.2.3. In the diagram in Figure 2, containments 1, 2 and 4 are proper, the group C^{0}(B_{n}(R)) is normal in all groups that contain it and the rectangle at the lower right is a lattice. There are no PL elements in Aut_{LI}^{0}() − C^{0}(), but there are PL elements in Aut_{π}^{0}() − Aut_{LI}^{0}() (at least when n is not a prime power).

Proof. That containments 1 and 4 hold is clear, and the other two follow from Lemma 5.3.3(e) and Corollary 5.3.5.2. Containment 1 is proper by Theorem 4.4.3, 2 is proper by Example 5.3.4 and 4 is proper by Proposition 5.3.1 since lifts are non-trivial and fix at least one generator. That the lower right rectangle is a lattice follows from Lemma 5.3.3(c). The existence and non-existence of PL elements is discussed in Example 5.3.4 and Lemma 5.3.3(e).

We can say something about structure. For 0 ≤ i < n − 1, let Out_{I_{i}}^{0}(B_{n}(R)) be the image of Aut_{I_{i}}^{0}(F_{n,∞}, t_{n−1}) under the map Aut(F_{n,∞}, t_{n−1}) → Out_{o}(B_{n}(R)), and let Out_{LI}^{0}(B_{n}(R)) be the image of Aut_{LI}^{0}(F_{n,∞}, t_{n−1}). To denote the elements of Aut_{LI}^{0}(F_{n,∞}, t_{n−1}) that fix g_{i}, we will use Aut_{LI∩g_{i}}^{0}(F_{n,∞}, t_{n−1})

Proposition 6.2.4. Let 0 ≤ i < n − 1.

(a) The sequence

1 → C^{0}(B_{n}(R), t_{n−1}) → Aut_{I_{i}}^{0}(F_{n,∞}, t_{n−1}) → Out_{I_{i}}^{0}(B_{n}(R)) → 1

is split exact with section \xymatrix{Out_{I_{i}}^{0}(B_{n}(R)) \ar[r]^{∼} & Aut_{g_{i}}^{0}(F_{n,∞}, t_{n−1})} , so

Aut_{I_{i}}^{0}(F_{n,∞}, t_{n−1}) ≃ F_{n,[0,n−1]} ⋊ Aut_{g_{i}}^{0}(F_{n,∞}, t_{n−1})

(b) The sequence

1 → C^{0}(B_{n}(R), t_{n−1}) → Aut_{LI}^{0}(F_{n,∞}, t_{n−1}) → Out_{LI}^{0}(B_{n}(R)) → 1

is split exact with a section \xymatrix{Out_{LI}^{0}(B_{n}(R)) \ar[r]^{∼} & Aut_{LI∩g_{i}}^{0}(F_{n,∞}, t_{n−1})} for each i. So

Aut_{LI}^{0}(F_{n,∞}, t_{n−1}) ≃ F_{n,[0,n−1]} ⋊ Out_{LI}^{0}(B_{n}(R))

(c) Out_{I_{i}}^{0}(B_{n}(R)) and Out_{LI}^{0}(B_{n}(R)) are torsion free.

Proof. (a) follows from Lemma 5.3.3(c) and remarks in Definition 3.1.1.

(b) This follows immediately from (a), since conjugation by w ∈ C^{0}(B_{n}(R), t_{n−1}) is locally inner.

(c) follows from the splittings in (a) and (b) because a non-trivial, orientation preserving, self homeomorphism of R has infinite order.

6.3. Results relating two values (m < n). For 2 ≤ m < n there is an embedding τ_{m,n} of F_{m,∞} into F_{n,∞} taking t_{m−1} to t_{n−1}, that extends to an embedding Θ_{m,n} of Aut^{0}(F_{m,∞}, t_{m−1}) into Aut^{0}(F_{n,∞}, t_{n−1}). We call the map Θ_{m,n} the lift from m to n. The lift Θ_{m,n} also extends the embedding of Π_{m} into Π_{n}. Moreover, locally inner automorphisms are lifted to locally inner automorphisms.

Theorem 6.3.1. Let 2 ≤ m < n be integers.

(a) There is a unique monomorphism of group pairs τ_{m,n} : (F_{m,∞}, t_{m−1}) → (F_{n,∞}, t_{n−1}) so that (g_{m,i})τ_{m,n} = g_{n,i} for 0 ≤ i ≤ (m − 2).

(b) there is a monomorphism Θ_{m,n} : Aut^{0}(F_{m,∞}, t_{m−1}) → Aut^{0}(F_{n,∞}, t_{n−1}) so that if θ ∈ Aut^{0}(F_{m,∞}, t_{m−1}) then θ′ = θΘ_{m,n} is the only map that makes the following diagram commute.

\xymatrix{(F_{m,∞}, t_{m−1}) \ar[d]^{θ} \ar[r]^{τ_{m,n}} & (F_{n,∞}, t_{n−1}) \ar[d]^{θ′} & (⟨g_{m−1}, …, g_{n−2}, t_{n−1}⟩, t_{n−1}) \ar[d]^{1} \ar@{^{(}->}[l] \\ (F_{m,∞}, t_{m−1}) \ar[r]^{τ_{m,n}} & (F_{n,∞}, t_{n−1}) & (⟨g_{m−1}, …, g_{n−2}, t_{n−1}⟩, t_{n−1}) \ar@{^{(}->}[l]}

(c) The lift Θ_{m,n} preserves permutations, i.e., the following diagram commutes.

\xymatrix{Aut^{0}(F_{m,∞}, t_{m−1}) \ar[r]^{Θ_{m,n}} \ar[d]^{π_{m}} & Aut^{0}(F_{n,∞}, t_{n−1}) \ar[d]^{π_{n}} \\ Π_{m} \ar@{^{(}->}[r] & Π_{n}}

(d) The lift Θ_{m,n} maps Aut_{LI}^{0}(F_{m,∞}, t_{m−1}) to Aut_{LI}^{0}(F_{n,∞}, t_{n−1}). The map τ_{m,n} maps F_{m,[0,m−1]} to F_{n,[0,n−1]}. If we define Υ_{m,n} : F_{m,[0,m−1]}^{m−1} → F_{n,[0,n−1]}^{n−1} by (W_{0}, …, W_{m−2})Υ_{m,n} = (W_{0}τ_{m,n}, …, W_{m−2}τ_{m,n}, 1, …, 1) then the following diagram commutes.

\xymatrix{Aut_{LI}^{0}(F_{m,∞}, t_{m−1}) \ar[r]^{Θ_{m,n}} \ar[d]^{ι_{m}} & Aut_{LI}^{0}(F_{n,∞}, t_{n−1}) \ar[d]^{ι_{n}} \\ F_{m,[0,m−1]}^{m−1} \ar[r]^{Υ_{m,n}} & F_{n,[0,n−1]}^{n−1}}

Note that the maps ι_{m}, ι_{n} are not homomorphisms.

Proof. (a), (b), and (c) are just Proposition 4.1.2, Theorem 4.1.6, and Proposition 4.4.2 respectively.

(d) The statement about Θ_{m,n} is Corollary 5.3.5.1. The statement about τ_{m,n} follows from Lemma 4.1.5. Commutativity follows from Lemma 5.3.5.

In contrast with Theorem 6.3.1 part (b), standard lifts do not normalize A_{n}(R), as we get from Theorem 5.4.7.

Theorem 6.3.2. (a) For integer values 2 ≤ m < n − 1, there is an isomorphic copy of Aut^{0}(F_{m,∞}, t_{m−1}) in Aut(B_{n}(R)) that has trivial intersection with Aut(A_{n}(R)). (b) For integer values 2 ≤ m = n − 1, there is an isomorphic copy of Aut_{LI}^{0}(F_{m,∞}, t_{m−1}) in Aut(B_{n}(R)) that has trivial intersection with Aut(A_{n}(R)).

However, when m < n are such that (m − 1)|(n − 1), we can do symmetric lifts from m to n. For α ∈ Aut^{0}(F_{m,∞}, t_{m−1}), the symmetric lift αΛ_{m,n} can be thought of as the composition of the standard lift αΘ_{m,n} with its iterated (m − 1)- step rotations. The lift αΛ_{m,n} normalizes F_{n,∞} and commutes with t_{m−1}, but in general it does not normalize A_{n}(R). In the particular case m = 2 we get that symmetric lifts from Aut^{0}(F_{2,∞}, t_{1}), do normalize A_{n}(R). From Theorems 4.3.1 and 5.4.8 we get the following.

Theorem 6.3.3. Let 2 ≤ m < n be such that (m − 1)|(n − 1).

(a) There is a monomorphism Λ_{n} : Aut^{0}(F_{m,∞}, t_{m−1}) → Aut^{0}(F_{n,∞}, t_{m−1}) so that if θ ∈ Aut^{0}(F_{m,∞}, t_{m−1}), then θ′ = θΛ_{m,n} is the only homomorphism in Aut^{0}(F_{n,∞}, t_{m−1}) that makes the following diagram commute.

\xymatrix{(F_{m,∞}, t_{m−1}) \ar[d]^{θ} \ar[r]^{τ_{m,n}} & (F_{n,∞}, t_{n−1}) \ar[d]^{θ′} \\ (F_{m,∞}, t_{m−1}) \ar[r]^{τ_{m,n}} & (F_{n,∞}, t_{n−1})}

In particular if θ ∈ Aut^{0}(F_{2,∞}, t_{1}) then θΛ_{2,n} normalizes A_{n}(R).

(b) If m > 2 and θ ∈ Aut(A_{m}(R)) is conjugation by 1 ≠ w ∈ C^{0}(B_{m}(R), t_{m−1}) then θΛ_{m,n} does not normalize A_{n}(R).

7. Questions

We gather some questions raised by the material in the paper. Recall the graded groups 𝒜 and ℒ𝒞 defined in Section 5.3. We introduce two more graded subgroups of 𝒜 to hang questions on. We let ℒℛ𝒞 be the smallest graded subgroup of 𝒜 that contains all the C^{0}(B_{n}(R), t_{n−1}) and that is closed under lifts and rotations. Note that 𝒜 is closed under rotations. We let ℒℛ𝒫ℒ be the smallest graded subgroup of 𝒜 that contains all the Aut_{P L}^{0}(F_{n,∞}, t_{n−1}) and that is closed under lifts and rotations.

Question 7.1. Is ℒℛ𝒫ℒ = 𝒜? This asks if we have found all automorphisms of B_{n}(R).

Question 7.2. Is ℒℛ𝒞 = ℒℛ𝒫ℒ ?

Question 7.3. Is ℒ𝒞 = ℒℛ𝒞 ?

Example 5.3.4 shows that ℒ𝒞 is a proper subset of ℒℛ𝒫ℒ , so at least one of Question 7.2 or 7.3 has answer no.

Question 7.4. Is containment 3 in the diagram above Proposition 6.2.3 proper?

To answer at least some of the questions above, it seems that techniques will have to be developed that go beyond the techniques of the current paper.

Question 7.5. Is Aut_{LI}^{0}(F_{n,∞}, t_{n−1}) preserved by rotations? Rotations preserve Aut^{0}(F_{n,∞}, t_{n−1}) by Lemma 4.2.1.

Question 7.6. Is there torsion in Out(B_{3}(R))?

Question 7.7. Is there torsion in Out_{π}(B_{n}(R))?

The example of torsion that we construct in Section 5.5 has a non-trivial permutation and only works for n ≥ 4. Is there an example of torsion that does not permute the residues?

Question 7.8. Is there a bound on the torsion in Out(B_{n}(R))? Even if non-trivial permutations have to be involved, there are relevant permutations with order much higher than that of our example.

Question 7.9. What is the image of Aut(A_{n}) under π_{n}? See Theorem 6.1.8(c).

Question 7.10. Is Aut(A_{n}(R)) normal in Aut(B_{n}(R))? Is the automorphism tower of A_{n}(R) contained in Aut(B_{n}(R))? Is Aut^{2}(A_{n}(R)) = Aut(A_{n}(R))? These questions ask about increasingly strict conditions on Aut(A_{n}(R)). There is no particular reason to suspect that even the first question has an affirmative answer.

Question 7.11. For 2 < m < n with (m − 1) | (n − 1), does image of Λ_{m,n} defined in Theorem 4.3.1 have trivial intersection with Aut(A_{n}(R))? Theorem 5.4.8 shows that the image of the inner automorphisms of A_{m}(R) under Λ_{m,n} has trvial intersection with Aut(A_{n}(R)).

Question 7.12. Does the image of Λ_{2,n} contain a representative of every element of Out_{o}(A_{n}(R))? This asks if we have found all automorphisms of A_{n}(R). This breaks down into two parts. Is the image of Λ_{2,n} equal to all of Aut^{0}(F_{n,∞}, t_{1}) modulo C^{0}(B_{n}(R), t_{n−1})? Is there are representative of every element of Out_{o}(A_{n}(R)) in Aut^{0}(F_{n,∞}, t_{1})? There is no reason to expect either question to have an affirmative answer.

The authors have thought about the above questions at least to some extent, and have not pursued such obvious questions as: what is the full structure of the automorphism and outer automorphism groups, do they have free subgroups, are any of them finitely generated or finitely presented, etc.

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