QUASI-ISOMETRICALLY EMBEDDED SUBGROUPS

OF THOMPSON’S GROUP F

José Burillo

Abstract. The goal of this paper is to construct quasi-isometrically embedded subgroups of Thompson’s group F which are isomorphic to F × ℤ^{n} for all n. A result estimating the norm of an element of Thompson’s group is found. As a corollary, Thompson’s group is seen to be an example of a finitely presented group which has an infinite-dimensional asymptotic cone.

The interesting properties of Thompson’s group F have made it a favorite object of study among group theorists and topologists. It was first used by McKenzie and Thompson to construct finitely presented groups with unsolvable word problems ([5]). It is of interest also in homotopy theory in work related to homotopy idempotents, due to its universal conjugacy idempotent map ϕ, also used to see that F is an infinitely iterated HNN extension. In [1] Brown and Geoghegan found F to be the first torsion-free infinite-dimensional F P_{∞} group. Also, F contains an abelian free group of infinite rank, but it does not admit a free non-abelian subgroup.

Many questions about F are still open, in particular it is not known whether F is automatic, or what is its Dehn function —although some estimates have been found by Guba, who proves it is polynomial in [4]—. The amenability of F is also unknown, fact that is of considerable interest since both the affirmative or negative answer would provide counterexamples to open questions (see [2]).

Questions about the geometric properties of F have also been proposed. Bridson raised the question of whether F and F × ℤ could be quasi-isometric. In this paper we provide a partial answer to this question, proving that F admits a quasiisometrically embedded subgroup isomorphic to F × ℤ. As a corollary, F is the first example of a finitely presented group whose asymptotic cone is infinite-dimensional (see [3]).

There are in the literature several interpretations of F that are useful to study it. Cannon, Floyd and Parry provide two of these interpretations, one as a group of homeomorphisms of the interval [0, 1], and another as a group of isomorphisms of rooted binary trees. In [1] Brown and Geoghegan prove that F is isomorphic to certain group of piecewise linear homeomorphisms of ℝ, fact that will be used extensively in this paper. This construction allows us to translate group-theoretical questions to the setting of these homeomorphisms of ℝ.

The organization of this paper is as follows: after a brief summary of results about F in section 1, an estimate of the norm of an element of F is found in section 2. The last section is dedicated to state and prove the results about the different subgroups of F.

The author would like to thank Z. Nitecki and J. Taback for useful comments in the development of this work.

1. Generalities on Thompson’s group F

Thompson’s group F is the group defined by the following infinite presentation:

P = ⟨x_{k}, k ≥ 0 | x_{i}^{−1}x_{j}x_{i} = x_{j+1}, if i < j⟩ .

Even though this is an infinite presentation, it follows from the relations that the generators x_{k}, for k > 1, are consequence of x_{0} and x_{1}. In fact, F admits a finite presentation given by

F = ⟨x_{0}, x_{1} | [x_{0}x_{1}^{−1}, x_{0}^{−1}x_{1}x_{0}], [x_{0}x_{1}^{−1}, x_{0}^{−2}x_{1}x_{0}^{2}]⟩

(see [1]). Throughout this paper, every time we refer to the word metric, or the norm, or the distance in F, it will make reference to the finite presentation F of F.

It is a consequence of the presentation P that the map ϕ : F ⟶ F defined by ϕ(x_{i}) = x_{i+1} is a conjugacy idempotent, i.e., satisfies ϕ^{2}(x) = x_{0}^{−1}ϕ(x)x_{0}, for all x ∈ F. Also, ϕ is injective, mapping F to the copy of itself generated by x_{k}, for k ≥ 1. Moreover, this map is shown in [1] to be universal among conjugacy idempotents.

It is also seen in [1] that the elements of F admit a unique normal form in the generators of P. From the relations in P it is easy to see that any element of F admits an expression of the form

x_{i_{1}}^{r_{1}}x_{i_{2}}^{r_{2}} … x_{i_{n}}^{r_{n}}x_{j_{m}}^{−s_{m}} … x_{j_{2}}^{−s_{2}}x_{j_{1}}^{−s_{1}}

such that (1) r_{1}, …, r_{n}, s_{1}, …, s_{m} > 0,

(2) i_{1} < i_{2} < … < i_{n},

(3) j_{1} < j_{2} < … < j_{m}, and

(4) i_{n} ≠ j_{m}. This expression is not unique if we do not require an extra condition: if for some i both x_{i} and x_{i}^{−1} appear, then either x_{i+1} or x_{i+1}^{−1} must appear as well. For otherwise there would be a subproduct of the form x_{i}ϕ^{2}(x)x_{i}^{−1} that could be replaced by ϕ(x). From this construction it is clear that, given a word in the generators x_{k} of P, using the relations and the extra condition, we can obtain the unique normal form, and the length of the word does not increase in this process. In other words, the unique normal form is the shortest of all the words that represent a given element in the generators of P. This fact will be used later in this paper.

Brown and Geoghegan show in [1] that G admits an isomorphism with the group of certain piecewise linear homeomorphisms of ℝ. Let

f_{k} : ℝ ⟶ ℝ

be the map defined by

f_{k}(t) = \{ \begin{matrix}t & if t ≤ k, \\ 2t − k & if k ≤ t ≤ k + 1, \\ t + 1 & if t ≥ k + 1. \end{matrix}

The group G generated by the maps f_{k}, for all integers k ≥ 0, and with right action, is isomorphic to Thompson’s group, with x_{k} identified with f_{k}. The right action means that the composition of two maps is written the opposite way: the element x_{i}x_{j} of F is associated with the element f_{i}f_{j} of G, which represents the map f_{j} ∘ f_{i} on ℝ. From now on we will identify the groups F and G. This construction will be extremely useful in section 2.

Using this construction it is easy to see more properties of F: the subgroup of F generated by the elements x_{2k}x_{2k+1}^{−1}, for k ≥ 0, is a free abelian subgroup of infinite rank. To see that two of these elements commute, observe that the map f_{2n}f_{2n+1}^{−1} is the identity except in the interval [2n, 2n + 2]. Also, due to this fact, it is clear that x_{0}x_{1}^{−1} commutes with x_{k} for k ≥ 2, so the subgroup generated by x_{0}x_{1}^{−1}, x_{2} and x_{3} is isomorphic to F × ℤ. In section 3 it will be proved that this subgroup is nondistorted in F.

For an exhaustive survey of the properties of Thompson’s groups (not only F) and their geometric interpretations, see [2].

2. The estimate of the norm

The geometric interpretation given by the maps f_{k} provides a method to compute the word metric of F (with respect to the finite presentation F). For instance, to compute the norm of an element x of F, we can study the corresponding map f in G: we know that f can be obtained as a composition of the maps f_{0} and f_{1} and their inverses, and we only need to estimate how many occurrences of f_{0} and f_{1} and their inverses we need to obtain f. This can be studied from properties of the graph of f.

Given a point (a, b) of the graph of f, with b = f(a), since f is piecewise linear, we denote by f′_{+}(a) and f′_{−}(a) the right and left derivatives of f at a. If f′_{+}(a) ≠ f′_{−}(a), we say that the point (a, b) is a breaking point of the graph of f.

To completely understand the maps in G we need to study how multiplying by a generator affects a map. Let f ∈ G be one of these piecewise linear homeomorphisms. Then, the map ff_{i} = f_{i} ∘ f has a graph that can be easily related to the graph of f. Since the map f_{i} has slope 2 only on those points with y-coordinate in the interval [i, i + 2], the graph of ff_{i} is obtained by stretching the portion of the graph that has y-coordinate in [i, i + 1] to the interval [i, i + 2], and all the graph is moved one unit up in all the points with y ≥ i + 1. A point (a, b) on the graph of f with b ∈ (i, i + 1) appears now as the point (a, i + 2(b − i)), and the derivatives satisfy

(ff_{i})′_{+}(a) = 2f′_{+}(a)  and  (ff_{i})′_{−}(a) = 2f′_{−}(a).

Similarly, the map ff_{i}^{−1} shrinks the interval [i, i + 2] down to [i, i + 1], and the derivatives get divided by 2. We will use this facts in the proof of the norm estimate below.

The following lemma is an example of how this maps can be used to obtain group-theoretical properties:

Lemma 1. Let f ∈ G, and let (a, b) be a point of the graph of f. Assume that one of the two derivatives f′_{+}(a) and f′_{−}(a) is different from 1. Then the norm of f can be bounded by

|f|_{G} ≥ max{1, a − 2, b − 2}.

In particular this applies to any breaking point of the graph of f.

Proof. Since |f|_{G} = |f^{−1}|_{G}, we only need to prove that |f|_{G} ≥ b − 2. And clearly we can assume b ≥ 3.

Observe that in f_{0} and f_{1} the highest point with a derivative different from 1 is the point (2, 3) in f_{1}, and further compositions with either f_{0} or f_{1} can only increase the y-coordinate by 1, and double the slope only at a point with y-coordinate in [0, 3]. So to achieve a derivative different from 1 in (a, b) one can start with f_{1} and compose it with b − 3 generators more, at least. So one needs to compose at least b − 2 generators to obtain a graph that has a derivative different from 1 in (a, b).□

The next result is the estimate of the norm of an element of G in terms of the unique normal form.

Proposition 2. Let f ∈ G be an element with normal form

f = f_{i_{1}}^{r_{1}} … f_{i_{n}}^{r_{n}}f_{j_{m}}^{−s_{m}} … f_{j_{1}}^{−s_{1}}.

Let D = r_{1} + … + r_{n} + s_{1} + … + s_{m} + i_{n} + j_{m}. Then

\frac{D}{6} − 2 ≤ |f|_{G} ≤ 3D.

Proof. Since |f|_{G} = |f^{−1}|_{G}, we can assume that i_{n} > j_{m}.

One of the inequalities is easy: rewrite the normal form in terms of f_{0} and f_{1} using f_{i} = f_{0}^{−i+1}f_{1}f_{0}^{i−1} to obtain a word representing f with only f_{0}, f_{1} and their inverses. It is easy to see that the length of this word is less than 3D.

It is also not difficult to see that |f|_{G} ≥ r_{1} + … + r_{n} + s_{1} + … + s_{m}: if |f|_{G} < r_{1} + … + r_{n} + s_{1} + … + s_{m}, there exists a word on f_{0} and f_{1} (and their inverses) that has length less than the unique normal form, contradicting the fact that the normal form is the shortest word for f. One could take this word and construct from it a normal form that would be shorter than the unique one.

The last step in the proof is to prove that

|f|_{G} ≥ \frac{i_{n}}{2} − 2.

Assume that r_{1} + … + r_{n} + s_{1} + … + s_{m} < i_{n}/2. If not, the inequality follows from the previous paragraph. Consider the graph of the element

g = f_{i_{1}}^{r_{1}} … f_{i_{n}}^{r_{n}},

the positive part of the normal form. Each one of these generators performs a stretching of the graph (see above), the last one being a stretching of the interval [i_{n}, i_{n} + 1] into [i_{n}, i_{n} + 2]. So in the graph of g there is a point P with coordinates (x, i_{n} +1) such that the two derivatives at this point are equal to 2^{N} where N ≥ r_{n}. We want to follow the movement down of P after composing with all the inverses that appear in the normal form. The desired conclusion is that at the end, in the graph of f, the point that corresponds to P is a breaking point, or else the function f still has derivatives 2^{N} at this point.

Observe the effect that composing with f_{j_{m}}^{−1} has on P. If j_{m} = i_{n} − 1, in gf_{j_{m}}^{−1} the point corresponding to P is now a breaking point: its derivatives are 2^{N} and 2^{N−1}. Further compositions with any f_{i}^{−1} will keep this point a breaking point, since i_{n} > j_{m} > … > j_{1}. If j_{m} < i_{n} − 1, then in gf_{j_{m}}^{−1} the point P has just seen its y-coordinate decreased by one and the derivatives are still both 2^{N}.

In the graph of gf_{j_{m}}^{−s_{m}}, the situation of the point P depends on the value of s_{m}:

(1) If s_{m} < i_{n} − j_{m} − 1, P is not a breaking point and the derivatives are both 2^{N}.

(2) If s_{m} = i_{n} − j_{m} − 1, the last composition by f_{j_{m}}^{−1} has made P a breaking point.

(3) If s_{m} > i_{n} − j_{m} − 1, P is now a breaking point whose y-coordinate is not an integer, and in any case it will remain a breaking point throughout, so it will be a breaking point of f.

The key to this argument is to observe that since i_{n} > j_{m} > … > j_{1}, a composition by an f_{i}^{−1} cannot decrease the right derivative at P without decreasing the left derivative. One of these compositions either divides the left derivative by 2 without touching the right derivative, or it divides both derivatives by 2. And it divides both derivatives by 2 only after P has been already made a breaking point. So the conclusion is that in f, either the derivatives at this point are still 2^{N} (if the s_{1}, …, s_{m} are small enough) or else it is a breaking point. In any case one of the two derivatives at this point is not 1. We need now to compute the y-coordinate of this point to apply Lemma 1.

In g the y-coordinate of P was i_{n} + 1. Every application of an f_{i}^{−1} may decrease the y-coordinate at most by 1. So the y-coordinate is at least

i_{n} + 1 − s_{1} − … − s_{m},

but from our assumption, r_{1} + … + r_{n} + s_{1} + … + s_{m} ≤ i_{n}/2, we conclude that the y-coordinate of this point is at least i_{n}/2. From Lemma 1 it follows now that

|f|_{G} ≥ \frac{i_{n}}{2} − 2.

Combining all inequalities (including i_{n} > j_{m}) we have

|f|_{G} ≥ max \{r_{1} + … + r_{n} + s_{1} + … + s_{m}, \frac{i_{n}}{2} − 2, \frac{j_{m}}{2} − 2\}

≥ \frac{r_{1} + … + r_{n} + s_{1} + … + s_{m} + \frac{i_{n}}{2} + \frac{j_{m}}{2} − 4}{3}

≥ \frac{D}{6} − 2,

and the proof is complete. □

3. Quasi-isometrically embedded subgroups

Recall that a map

F : X ⟶ Y

between metric spaces is called a quasi-isometric embedding if there exist constants K, C > 0 such that

\frac{d(x, x′)}{K} − C ≤ d(F (x), F (x′)) ≤ Kd(x, x′) + C,

for all x, x′ ∈ X. If G is a finitely generated group, and H is a finitely generated subgroup, then the fact that the inclusion is a quasi-isometric embedding is equivalent to say that the distortion function

h(r) = \frac{1}{r} max \{|x|_{H} | x ∈ H, |x|_{G} ≤ r\}

is bounded. If a subgroup is quasi-isometrically embedded, then its own word metric is equivalent to the metric induced by the word metric of the ambient group, and the distortion is bounded.

Our goal is to prove that several subgroups of F are quasi-isometrically embedded. The first one and from which all the other ones will be deduced, is the subgroup isomorphic to F × ℤ generated by the elements x_{0}x_{1}^{−1}, x_{2} and x_{3}.

Theorem 3. The map

Φ : F × ℤ ⟶ F

defined by

Φ(x, t^{k}) = (x_{0}x_{1}^{−1})^{k}ϕ^{2}(x)

QUASI-ISOMETRICALLY EMBEDDED SUBGROUPS OF THOMPSON’S GROUP F 7

where x ∈ F and t is the generator of ℤ, is a quasi-isometric embedding. Proof. We need to prove that there exist constants K, C > 0 such that

\frac{1}{K} |(x, t^{k})|_{F ×ℤ} − C ≤ |Φ(x, t^{k})|_{F} ≤ K|(x, t^{k})|_{F ×ℤ} + C.

Assume x ∈ F has normal form

x_{i_{1}}^{r_{1}} … x_{i_{n}}^{r_{n}}x_{j_{m}}^{−s_{m}} … x_{j_{1}}^{−s_{1}},

and let

D = r_{1} + … + r_{n} + s_{1} + … + s_{m} + i_{n} + j_{m}.

Then

|(x, t^{k})|_{F ×ℤ} = |k| + |x|_{F} ,

and, by the norm estimate,

|k| + \frac{D}{6} − 2 ≤ |(x, t^{k})|_{F ×ℤ} ≤ |k| + 3D.

To compute the normal form of the element Φ(x, t^{k}) ∈ F, assume that k ≥ 0. Observe that when k < 0, we can just compute the norm of Φ(x, t^{k})^{−1} = ϕ^{2}(x)^{−1}(x_{0}x_{1}^{−1})^{−k} = (x_{0}x_{1}^{−1})^{−k}ϕ^{2}(x^{−1}). The normal form is:

(x_{0}x_{1}^{−1})^{k}ϕ^{2}(x) = (x_{0}x_{1}^{−1})^{k}x_{i_{1}+2}^{r_{1}} … x_{i_{n}+2}^{r_{n}}x_{j_{m}+2}^{−s_{m}} … x_{j_{1}+2}^{−s_{1}}
= x_{0}^{k}x_{i_{1}+k+2}^{r_{1}} … x_{i_{n}+k+2}^{r_{n}}x_{j_{m}+k+2}^{−s_{m}} … x_{j_{1}+k+2}^{−s_{1}}x_{k}^{−1}x_{k−1}^{−1} … x_{2}^{−1}x_{1}^{−1}.

Then,

\frac{D + 4k + 4}{6} − 2 ≤ |Φ(x, t^{k})|_{F} ≤ 3(D + 4k + 4).

Combining these inequalities with the ones before, we obtain:

\frac{|(x, t^{k})|_{F ×ℤ}}{18} − 2 ≤ |Φ(x, t^{k})|_{F} ≤ 18|(x, t^{k})|_{F ×ℤ} + 48.

which gives the desired result. □

Once we know that F × ℤ quasi-isometrically embeds in F, we can extend this result to other subgroups: Corollary 4. Ths subgroup generated by the elements

x_{0}x_{1}^{−1}, x_{2}x_{3}^{−1}, …, x_{2n−2}x_{2n−1}^{−1}, x_{2n}, x_{2n+1}

is isomorphic to F × ℤ^{n} and it is quasi-isometrically embedded.

Also, the free abelian subgroup ℤ^{n} in F × ℤ^{n} is also quasi-isometrically embedded. This implies that the asymptotic cone of F contains copies of ℝ^{n} for arbitrary n. For definitions and properties of asymptotic cones see [3]. Corollary 5. The asymptotic cone of F is infinite-dimensional.

References

1. Brown, K.S., Geoghegan. R., An infinite-dimensional torsion-free F P_{∞} group, Invent. Math. 77 (1984), 367–381.

2. Cannon, J.W., Floyd, W.J., Parry, W.R., Introductory notes on Richard Thompson’s groups, L’Ens. Math. 42 (1996), 215–256.

3. Gromov, M., Asymptotic invariants of infinite groups, Geometric Group Theory II (G. Niblo, M. Roller, eds.), London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1993.

4. Guba, V., Polynomial upper bounds for the Dehn function of R. Thompson’s group F, preprint.

5. McKenzie, R., Thompson, R.J., An elementary construction of unsolvable word problems in group theory, Word Problems (W.W. Boone, F.B. Cannonito, R.C. Lyndon, eds.), Studies in Logic and the Foundation of Mathematics, vol.71, North–Holland, Amsterdam, 1973, pp. 457– 478.

Dept. of Mathematics, Tufts University, Medford, MA 02155, U.S.A. E-mail address: jburillo@emerald.tufts.edu