The Tits Alternative for Out(F_{n}) III: Solvable Subgroups

Mladen Bestvina, Mark Feighn, and Michael Handel

February 11, 1997

Contents

1 Introduction 1

2 Notation and Preliminaries 3 2.1 Lifts to C_{∞} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Relative train track maps . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Property A 6

4 Abelian subgroups are finitely generated 11

5 Proof of Proposition 1.2 15

6 Proof of Theorem 1.1 19

1 Introduction

Let F_{n} denote a free group of rank n. The group Out(F_{n}) contains mapping class groups of compact surfaces and maps onto GL(n, ℤ). It is perhaps not surprising that Out(F_{n}) behaves at times like a mapping class group and at times like a linear group. J. Birman, A. Lubotzky, and J. McCarthy [BLM83] showed that solvable subgroups of mapping class groups are finitely generated and virtually abelian. Of course, GL(3, ℤ) contains the Heisenberg group which is solvable but not virtually abelian. In this paper, we show that, with respect to the nature of solvable subgroups, Out(F_{n}) behaves more like mapping class groups.

Theorem 1.1. Every solvable subgroup of Out(F_{n}) has a subgroup of index at most 3^{5n^{2}} that is finitely generated and free abelian.

The rank of an abelian subgroup of Out(F_{n}) is bounded by vcd(Out(F_{n})) = 2n − 3 for n > 1 [CV86].

Since Aut(F_{n}) embeds in Out(F_{n+1}), solvable subgroups of Aut(F_{n}) are also virtually abelian. The Theorem 1.1 complements [BFH96b] where we show that Out(F_{n}) satisfies the Tits Alternative, i.e. subgroups of Out(F_{n}) are either virtually solvable or contain a free group of rank 2.

H. Bass and A. Lubotzky [BL94] have shown that solvable subgroups of Out(F_{n}) are virtually polycyclic. In particular, they are finitely generated. We include an independent proof of this fact for completeness and because the ingredients of the proof are needed for the proof of Theorem 1.1.

The starting point for this paper is a short exact sequence from [BFH96a] and [BFH96b]. Begin with a solvable subgroup H of Out(F_{n}). After passing to a finite index subgroup we may assume that H acts trivially on H_{1}(F_{n}; ℤ/3ℤ). By Theorem 8.1** of [BFH96a] and Proposition 3.5** of [BFH96b] there is an exact sequence

1 → H_{0} → H →^{Ω} ℤ^{b} → 1

where H_{0} is UP G (defined below).

There are two parts to the proof of Theorem 1.1. First we show that H_{0} is abelian by constructing an embedding Φ : H_{0} → ℤ^{r}. In the second part we show that Φ extends to a homomorphism Φ′ : H → ℤ^{r}. The direct sum of Ω and Φ′ is then an embedding of H into ℤ^{b+r}.

Our approach is motivated by the special case that H is realized as a subgroup of the mapping class group of a compact surface S whose fundamental group is identified with F_{n}. The surface S decomposes into a union of annuli A_{1}, …, A_{r} and subsurfaces S_{i} of negative Euler characteristic; virtually every η ∈ H is represented by a homeomorphism f : S → S that restricts to a Dehn twist on each A_{j} and that preserves each S_{i}. If η ∈ H_{0} then each f|S_{i} is the identity. The j^{th} coordinate of the homomorphisms Φ and Φ′ is the amount of twisting that occurs across A_{j}. For a further discussion of the geometric case, see Example 3.2.

We study an outer automorphism η through its lifts η̂: C_{∞} → C_{∞} to the Cantor set at infinity C_{∞} or equivalently (see Subsection 2.1) through the automorphisms Φ : F_{n} → F_{n} that represent η. The C_{∞} point of view simplifies certain proofs because it allows us to consider fixed ‘directions’ in F_{n} that are not periodic and therefore do not come from fixed elements of F_{n}.

In the course of proving Theorem 1.1 we prove the following result which is of independent interest. We state it here in terms of automorphisms although we prove it in terms of C_{∞}.

Proposition 1.2. Every abelian subgroup H ⊂ Out(F_{n}) has a virtual lift H̃ ⊂ Aut(F_{n}). If γ is a non-trivial primitive element of F_{n} that is fixed, up to conjugacy, by each element of H then H̃ can be chosen so that each element of H̃ fixes γ.

The paper is organized as follows. In section 2 we establish notation and record known results for future reference. In sections 3 and 4 we prove that H is abelian and that the above exact sequence is virtually central. In section 5 we prove Proposition 1.2 and in section 6 we prove Theorem 1.1.

2 Notation and Preliminaries

2.1 Lifts to C_{∞}

We assume that F_{n} is identified with π_{1}(R_{n}, ∗) where R_{n} is the rose with n petals and vertex ∗. Let R̃_{n} be the universal cover of R_{n} and let ˜∗ be a preferred lift of ∗. The space of ends of R̃_{n} is a Cantor set that we denote C_{∞}.

A marked graph is a graph G with a preferred vertex v along with a homotopy equivalence τ : (R_{n}, ∗) → (G, v) that identifies π_{1}(G, v) with π_{1}(R_{n}, ∗) and so also with F_{n}. A preferred vertex ṽ in the universal cover Γ of G provides an identification of the group T of covering translations of Γ with π_{1}(G, v) and so with F_{n}.

Let E(Γ) be the space of ends of Γ. The marking homotopy equivalence lifts to an equivariant map τ̃ : (R̃_{n}, ˜∗) → (Γ, ṽ) that induces a homeomorphism from C_{∞} to E(Γ) (See for example section 3.2 of [BFH96a]). We use this homeomorphism to identify E(Γ) with C_{∞} and so for the rest of this paper refer to the space of ends of Γ as C_{∞}. The action of T on Γ extends to an action (of F_{n}) on C_{∞} by homeomorphisms. If T is non-trivial, then the endpoints of its axis are the only fixed points for the action of T on C_{∞}.

A homotopy equivalence f : G → G induces an outer automorphism of π_{1}(G, v) and so an outer automorphism η of F_{n}. Suppose that f̃: Γ → Γ is a lift of f : G → G. Then f̃T is a lift of f for each T ∈ T and there exists a unique T′ ∈ T such that f̃T = T′f̃. This defines an automophism T ↦ T′ of T ; the induced automorphism Φ : F_{n} → F_{n} represents η. As f̃ varies over all lifts of f, Φ varies over all automorphisms representing η. This defines an isomorphism between the group of lifts of f and the group of automorphisms representing η.

Under the identification of T with F_{n}, the subgroup T (f̃) ⊂ T of covering translations that commute with f̃ corresponds to the fixed subgroup F ix(Φ) ⊂ F_{n}. Since F ix(Φ) is quasiconvex and finitely generated, the closure in C_{∞} of the endpoints of axes of elements of T (f̃) is identified with the space of ends of F ix(Φ) [Coo87].

Each lift f̃: Γ → Γ extends to a homeomorphism f̂: C_{∞} → C_{∞} . If L ⊂ Γ is a line with endpoints P, Q ∈ C_{∞}, then define f̃_{#}(L) ⊂ Γ to be the line with endpoints f̂(P) and f̂(Q). If T ↦ T′ is the automorphism of T determined by f̃, and if L is the axis of T then f̃_{#}(L) is the axis of T′. Thus T ↦ T′ and the corresponding Φ : F_{n} → F_{n} are determined by f̂. Conversely, since the endpoints of axes are dense in C_{∞}, Φ determines f̂. We have now shown that f̂ is independent of the choice of f and Γ. We will sometimes write η̂ instead of f̂ where η ∈ Out(F_{n}) is the outer automorphism determined by f : G → G.

We conclude this subsection by recording some facts for future reference. Denote the space of F_{n}- equivariant homeomorphisms of C_{∞} by EH(C_{∞}). The preceding paragraph shows that Aut(F_{n}) embeds in EH(C_{∞}). If H̃ ⊂ Aut(F_{n}) is a lift of H ⊂ Out(F_{n}) then we denote the corresponding lift to EH(C_{∞}) by Ĥ. For any subgroup 𝔽 ⊂ F_{n}, the closure in C_{∞} of the endpoints of the axes of elements in 𝔽 is denoted by C(𝔽) and is naturally identified with the space of ends of 𝔽.

Lemma 2.1. Suppose that H is a subgroup of Out(F_{n}) and that 𝔽 is an H-invariant (up to conjugacy) subgroup of F_{n} that is its own normalizer. Then

1. There is a well-defined restriction H|𝔽 ⊂ Out(𝔽).

2. If H is UP G, then H|𝔽 is UP G.

3. If 𝔽 has rank at least two, then any lift H̃|𝔽 ⊂ Aut(𝔽) of H|𝔽 extends uniquely to a lift H̃ ⊂ Aut(F_{n}).

4. If every element of the lift Ĥ|𝔽 ⊂ EH(C(𝔽)) of H|𝔽 fixes at least three points then Ĥ|𝔽 extends uniquely to a lift Ĥ ⊂ EH(C_{∞})

Proof of Lemma 2.1 Since 𝔽 is its own normalizer, two automorphisms of F_{n} that preserve 𝔽 are conjugate by an element of F_{n} if and only if they are conjugate by an element of 𝔽. Part (1) follows immediately. Part (2) is Lemma 4.13 of [BFH96b]. If 𝔽 has rank at least two, then there are no non-trivial inner automorphisms that pointwise fix 𝔽. It follows that the restriction homomorphism from the subgroup of Aut(F_{n}) representing η ∈ Out(F_{n}) to the subgroup of Aut(𝔽) representing η|𝔽 is injective. Part (3) follows easily. For part (4), note that a non-trivial covering translation does not fix more than two points in C_{∞} and so two lifts η̂_{1} and η̂_{2} of η that agree on three points must be equal.

2.2 Relative train track maps

We study an element η ∈ Out(F_{n}) through its lifts η̂ : C_{∞} → C_{∞}. We analyze the η̂’s by representing η as a homotopy equivalence f : G → G of a marked graph with particularly nice properties and studying the corresponding f̂’s. In this subsection we recall some properties of f : G → G.

A filtered graph is a marked graph along with a filtration ∅ = G_{0} ⊂ G_{1} ⊂ ⋯ ⊂ G_{K} = G where each G_{i} is obtained from G_{i−1} by adding a single edge E_{i}. We reserve the words path and loop for immersions of the interval and the circle respectively. If ρ is a map of the interval or the circle into G or Γ, then [ρ] is the unique path or loop that is homotopic to ρ rel endpoints if any. We say that a homotopy equivalence f : G → G respects the filtration if each f(E_{i}) = E_{i}u_{i,f} for some loop u_{i,f} ⊂ G_{i−1}. Let F HE(G, V) be the group (Lemma 6.2 of [BFH96b]) of homotopy classes, relative to vertices, of filtration respecting homotopy equivalences of G. There is a natural map from F HE(G, V) to Out(F_{n}). We say that η ∈ Out(F_{n}) is UP G (for unipotent with polynomial growth) if it is in the image of F HE(G, V) for some G. We say that a subgroup of Out(F_{n}) is UP G if each of its elements is. The main theorem of [BFH96b] states that every UP G subgroup H_{0} lifts to a subgroup K of F HE(G) for some G. We say that K is a Kolchin representative of H_{0}.

In general we will use K to denote a subgroup of F HE(G, V).

If L̃ ⊂ Γ is a line and l is the highest parameter value for which L̃ crosses a lift of E_{l}, then define the highest edge splitting of L̃ = … σ̃_{−1} · σ̃_{0} · σ̃_{1} … by subdividing at the initial vertex of each lift of E_{l} that is crossed, in either direction, by L. We refer to the vertices that determine this decomposition as the splitting vertices of the highest edge splitting. If f̃: Γ → Γ is a lift of f ∈ F HE(G, V) and if L̃ is f̃_{#}-invariant, then Lemma 5.2*** of [BFH96a] implies that [f̃(σ̃_{j})] = σ̃_{j+r} for some r and all j. Roughly speaking, f̃ acts on L̃ by translating the highest edge splitting by r units.

We will need the following fixed point results.

Lemma 2.2. • Given F HE(G, V) and distinct P_{1}, P_{2}, P_{3} ∈ C_{∞}, there is a line L̃ in Γ connecting P_{i} to P_{j} for some 1 ≤ i < j ≤ 3 with the following property: If f ∈ F HE(G, V) and P_{1}, P_{2}, P_{3} ∈ F ix(f̂) then f̃ fixes each highest edge splitting vertex in L̃.

• If f ∈ F HE(G, V) and f̃: Γ → Γ is fixed point free, then f̂ fixes exactly two points.

Proof of Lemma 2.2 For the first item, let L̃_{i,j} ⊂ Γ be the line connecting P_{i} to P_{j} and let l_{i,j} be the highest parameter value for which L̃_{i,j} crosses a lift of E_{l_{i,j}}. Assuming without loss that l_{1,2} ≥ l_{1,3}, l_{2,3}, let L̃ = L̃_{1,2} and let ṽ be any highest edge splitting vertex of L̃. Suppose that f ∈ F HE(G, V) and that f̂ fixes each P_{i}. If f̃ does not fix ṽ then f̃ translates the highest edge splitting vertices of L̃ away from one endpoint of L̃, say P_{1}, and toward the other, P_{2}. It follows that the highest edge splitting of L̃ is bi-infinite and hence that l_{1} = l_{2} = l_{3}. This implies that f̃ translates the highest edge splitting vertices of L̃_{2,3} away from P_{3} and toward P_{2}. But now f̃ translates the highest edge splitting vertices of L̃_{1,3} away from P_{1} and away from P_{3}. This contradiction verifies the first item.

We assume now that f ∈ F HE(G, V) and that f̃ is fixed point free. It suffices to show that f̂ fixes at least two points. Implicit in Proposition 6.21 of [BFH96a] is the existence of a half-infinite ray R̃_{+} ⊂ Γ with highest edge splitting R̃_{+} = σ̃_{0} · σ̃_{1} … such that [f̃(σ̃_{j})] = σ̃_{j+r} for some r > 0 and all j ≥ 0. Let ṽ_{j} be the initial vertex of σ̃_{j}. Since f̃ restricts to a bijection of vertices, there are unique vertices ṽ_{j} so that f̃(ṽ_{j}) = ṽ_{j+1} for all j ∈ ℤ. Let σ̃_{j} be the path connecting ṽ_{j} to ṽ_{j+1} for j < 0 and note that [f̃(σ̃_{j})] = σ̃_{j+r} for all j. If l is the highest parameter value for which R̃_{+} crosses a lift of E_{l}, then each σ̃_{j} contains exactly one lift of E_{l} and these lifts are distinct for distinct j. (The j < 0 case follows from the j ≥ 0 case which holds by construction.) It follows that … σ̃_{−1} · σ̃_{0} · σ̃_{1} … is an embedded line whose endpoints are both fixed by f̂.

Lemma 2.3. Up to conjugation by covering translations, each η ∈ Out(F_{n}) has only finitely many lifts η̂: C_{∞} → C_{∞} whose fixed point set contains at least three points.

Proof of Lemma 2.3 Represent η by a homotopy equivalence f : G → G of some marked graph and let f̃: Γ → Γ be a lift whose extension f̂ fixes at least three points. Given P_{1}, P_{2}, P_{3} ∈ F ix(f̂), let ṽ ∈ Γ be the unique vertex contained in each of the lines connecting two of the P_{i}’s. The bounded cancellation lemma [Coo87] implies that there is a bound, depending on f but not on the choice of the lift f̃ or the choice of points P_{i}, to the length of the path σ̃ connecting ṽ to f̃(ṽ). In particular, the projected image σ takes on only finitely many values as we vary the lift f̃ and the choice of points P_{i}.

Suppose that f̃_{i}, i = 1, 2, are lifts of f and that there exist lifts ṽ_{i} of a vertex v and σ̃_{i} of a path σ such that σ̃_{i} is the path connecting ṽ_{i} to f̃_{i}(ṽ_{i}). The covering translation T : Γ → Γ that carries ṽ_{1} to ṽ_{2} satisfies f̃_{2}T = T f̃_{1} and so conjugates f̂_{1} to f̂_{2}.

3 Property A

In this section we prove that a finitely generated, solvable UP G subgroup H_{0} embeds in some ℤ^{r} and hence that every solvable UP G subgroup is free abelian. Some of the arguments proceed by induction on the skeleta G_{i} of a filtered graph G. For this reason, we do not assume in this section that K is a Kolchin representative of some H_{0} but only that K is a finitely generated solvable subgroup of F HE(G, V) with the following feature: if E_{i} is not a loop then some u_{i,f} is non-trivial. In particular, we do not assume that K injects into Out(π_{1}(G)) and we allow G to have valence one vertices. Note that if some u_{i,f} is non-trivial, then the terminal vertex of E_{i} is contained in a loop in G_{i−1} and so must have valence at least two in G_{i−1}. Thus if E_{i} is the first edge to contain a vertex v, then either E_{i} is a loop or E_{i} has v as initial vertex. In either case, v is the initial vertex of E_{i}.

The group of lifts f̃: Γ → Γ [respectively f̂: C_{∞} → C_{∞}] of elements of K has a natural projection to K. By an action of K on Γ [respectively C_{∞}] by lifts we mean a section of this projection. In other words, an action by lifts is an assignment f ↦ f̃ [respectively f ↦ f̂] that respects composition. Every action s of K on Γ by lifts determines an action ŝ of K on C_{∞} by lifts and vice-versa.

For each edge E_{i} ⊂ G choose, once and for all, a lift Ẽ_{i}^{∗} ⊂ Γ. Define s_{i}(f) : Γ → Γ to be the unique lift of f ∈ K that fixes the initial endpoint of Ẽ_{i}^{∗} and note that s_{i} is an action of K on Γ by lifts. Denote the terminal endpoints of E_{i} and Ẽ_{i}^{∗} by v_{i} and ṽ_{i} respectively.

Definition 3.1. If v_{i} ∈ G_{i−1}, denote the component of G_{i−1} that contains v_{i} by B_{i} and the (necessarily s_{i}(f)-invariant) copy of the universal cover of B_{i} that contains ṽ_{i} by Γ_{i−1}.

We say that K (and the choice of the Ẽ_{i}^{∗}’s) satisfies Property A (for abelian) if:

• For each i, either E_{i} is a component of G_{i} (in which case E_{i} is a loop and u_{i,f} is trivial for all f ∈ K) or ṽ_{i} is a highest edge splitting vertex in a line L̃_{i} ⊂ Γ_{i−1} that is s_{i}(f)_{#}-invariant for all f ∈ K.

• If L̃_{i} and L̃_{j} have the same projection in G, then L̃_{i} = L̃_{j} and ṽ_{i} = ṽ_{j}.

If L̃_{i} projects to an indivisible loop α_{i} and if some u_{i,f} is non-trivial, then we say (abusing notation slightly) that α_{i} is an essential axis and that E_{i} is an essential edge. The set of essential axes is denoted A(K) and the set of essential edges is denoted E(K). If the essential axis α is associated to m_{α} essential edges, then we say that m_{α} is the multiplicity of α. In the analogy with the mapping class group of a compact surface, A(K) is the set of reducing curves and K can be chosen so that each m_{α} = 1.

For each essential axis α, let ṽ_{α} be the preferred vertex of the preferred lift α̃ (= ṽ_{i} ∈ L̃_{i} for any L̃_{i} that projects to α), let T_{α} be an indivisible covering translation whose axis is α̃ and let s_{α}(f) be the lift of f ∈ K that fixes ṽ_{α}. Note that s_{α} defines an action of K on Γ by lifts. Note also that if we think of α as a closed path with both endpoints at the projected image v of ṽ, then [f(α)] = α. It follows that s_{α}(f)_{#}(α̃) = α̃ and hence that s_{α}(f) commutes with T_{α}.

Example 3.2. We set notation for this geometric example as shown below: M is the orientable genus two surface with one boundary component; A ⊂ M is an embedded non-peripheral annulus; S = M \ A; G ⊂ M is the embedded graph shown below with vertices v_{1} and v_{2} and edges E_{1}, …, E_{5}; H_{0} ≅ ℤ is the subgroup of the mapping class group of M generated by the Dehn twist across the annulus A; and K ≅ ℤ ⊂ F HE(G, V) is generated by E_{i} ↦ E_{i} for i ≠ 2 and E_{2} ↦ E_{2}E_{1}.

There is a single essential edge E_{2} with essential axis α = E_{1}. The universal cover of M is identified with a convex subset of the hyperbolic plane H in the Poincare disk model. Choose a lift L̃ of E_{1} and let Ã be the lift of A that contains L̃. On the left [respectively right] of Ã is a copy S̃_{1} [respectively S̃_{2}] of the universal cover of S. Given D ∈ H_{0}, let D̃_{i} : M̃ → M̃ be the lift that is the identity on S̃_{i}. Note that D̂_{i} fix the endpoints of S_{i} and that D̂_{1} and D̂_{2} are the only lifts of D that fix the endpoints of L̃ and at least one other point. Note also that if D is a Dehn twist of order k around A then D̃_{1} = T^{k}D̃_{2} where T is the indivisible covering translation that preserves L̃.

The lifts D̃_{1} and D̃_{2} have natural counterparts on the graph level. They are s_{α}(f) and s_{2}(f) respectively.

Note that π_{1}(S) is the subgroup of F_{n} generated by E_{2}E_{1}Ē_{2}, E_{3}E_{1}Ē_{3}, E_{4} and E_{5}. The group H_{0} can be extended to a non-UP G abelian group H by adding an element that restricts to a pseudo-Anosov homeomorphism on S. This can be represented by a relative train map f : G → G that agrees with D on E_{1} and E_{2} and that has a single exponentially growing stratum with edges E_{3}, E_{4} and E_{5}.

The following lemma justifies our notation.

Lemma 3.3. If K satisfies property A, then there is an injective homomorphism Φ_{K} : K → ℤ^{r} where r is the cardinality of E(K). In particular, K is free abelian.

Proof of Lemma 3.3 If E_{i} is an essential edge with essential axis α then s_{i}(f) and s_{α}(f) are lifts of f that commute with T_{α} and so differ by an iterate of T_{α}. Define a homomorpism ϕ_{K}^{i} : K → ℤ by s_{i}(f) = T_{α}^{ϕ_{K}^{i}(f)}s_{α}(f) and define Φ_{K} to be the product of the ϕ_{K}^{i}’s. Then f is in the kernel of Φ_{K} if and only if f(E_{i}) = E_{i} for each essential edge E_{i} if and only if u_{i,f} is trivial for each essential edge E_{i}.

If f ≠ identity, then there is a smallest parameter value i > 1 for which u_{i}(f) is non-trivial. Since f|G_{i−1} is the identity, s_{i}(f) restricts to a non-trivial covering translation of Γ_{i−1}. The line L̃_{i} ⊂ Γ_{i−1} is s_{i}(f)-invariant so must be the axis of that covering translation. Thus E_{i} ∈ E(K) and f is not in the kernel of Φ_{K}. This proves that Φ_{K} is injective.

The following lemma produces a pair of fixed points in C_{∞} or equivalently a fixed line in Γ.

Lemma 3.4. Suppose that ψ̂_{1}, …, ψ̂_{m} : C_{∞} → C_{∞} are lifts of elements of a finitely generated UPG subgroup. If the ψ̂_{j}’s commute, then ∩_{j=1}^{m}Fix(ψ̂_{j}) contains at least two points.

Proof of Lemma 3.4 Choose a Kolchin representative K of the UPG subgroup. There are elements f_{j} ∈ K and commuting lifts f̃_{j} : Γ → Γ so that f̂_{j} = ψ̂_{j} for j = 1, …, m.

If f̃_{1} is fixed point free then (Lemma 2.2) Fix(f̂_{1}) is a pair of points {P, Q}. Since f̂_{j} commutes with f̂_{1}, f̂_{j} setwise preserves {P, Q} and we need only show that f̃_{j} does not reverse the orientation on the line L̃ connecting P and Q. Let L̃ = … σ̃_{−1} · σ̃_{0} · σ̃_{1} … be highest edge splitting of L̃. If f̃_{j} reverses the orientation on L̃, then for some j, [f̃(σ̃_{j})] equals σ̃_{j} with its orientation reversed. The projected image σ_{j} determines a conjugacy class in F_{n} that is periodic but not fixed under the action of the outer automorphism determined by f. This contradicts Proposition 4.5 of [BFH96b].

Suppose next that F ix(f̃_{1}) ≠ ∅ and that the group T (f̃_{1}) of covering translations that commute with f̃_{1} is trivial. The fixed point set of f_{j}, and hence of f̃_{j}, is a union of vertices and edges. Since T (f̃_{1}) is trivial, each vertex v ∈ G has at most one lift ṽ ∈ F ix(f̃_{1}). Since f̃_{j} commutes with f̃_{1}, it preserves F ix(f̃_{1}). We conclude that each vertex in F ix(f̃_{1}) is fixed by each f̃_{j}. (Recall that f_{j} fixes each vertex in G.) The edges with initial vertex in F ix(f̃_{1}) project to distinct edges in G. Let Ẽ_{k} be the unique such edge with minimal k, let B′ be the component of G_{k−1} that contains the terminal endpoint of E_{k} and let Γ′_{k−1} ⊂ Γ be the copy of the universal cover of B′ that contains the terminal endpoint of Ẽ_{k}. Since each f_{j} maps an initial segment of E_{k} to an initial segment of E_{k}, Γ′_{k−1} is f̃_{j}-invariant for each j. By our choice of k, the restriction f̃_{1}|Γ′_{k−1} is fixed point free. We can now repeat the argument of the first case on the restriction of the f̃_{j}’s to Γ′_{k−1}.

Finally, suppose that T (f̃_{1}) is non- trivial. Identify T (f̃_{1}) with a subgroup 𝔽 of F_{n}. The space of ends of 𝔽 is the closure C(𝔽) ⊂ Fix(f̂_{1}) ⊂ C_{∞} of the endpoints of axes for elements of T (f̃_{1}). Since f̃_{j} commutes with f̃_{1}, the automorphism of T determined by f̃_{j} preserves T (f̃_{1}) and f̂_{j} preserves C(𝔽). By Lemma 2.1, the ψ_{j}|𝔽’s are contained in a UP G subgroup of Out(𝔽).

We argue by induction on m, the m = 1 case following from the fact that C(𝔽) contains at least two points. Suppose that m > 1. By the inductive hypothesis, there exist P, Q ∈ C(𝔽) that are fixed by the m − 1 maps f̂_{2}|C(𝔽), …, f̂_{m}|C(𝔽). Since f̂_{1}|C(𝔽) is the identity, P and Q are fixed by each f̂_{j}.

Corollary 3.5. Every finitely generated solvable UPG subgroup H_{0} has a Kolchin representative that satisfies property A; in particular, H_{0} is free abelian.

Proof of Corollary 3.5 Choose a Kolchin representative K for H_{0}. We work our way up the strata E_{i}, modifying K so that it satisfies property A. Denote the restriction of K to G_{i} by K_{i}. Since G_{1} is a single edge, K_{1} has property A; we may assume by induction that K_{i−1} satisfies property A and is therefore abelian.

If v_{i} ∉ G_{i−1}, then E_{i} is a component of G_{i} and K_{i} satisfies property A. We may therefore assume that Γ_{i−1} and the action ŝ_{i} of K on C_{∞} are defined. The ends of Γ_{i−1} define a subset C_{∞}^{∗} ⊂ C_{∞}.

Let R : K_{i} → K_{i−1} be the restriction homomorphism. If R is an isomorphism, then K_{i} is abelian. Choose generators g_{1}, …, g_{m} for K_{i}, representing ψ_{1}, …, ψ_{m} ∈ H_{0}, and let ψ̂_{j}^{∗} = ŝ_{i}(g_{j})|C_{∞}^{∗}. Note that the ψ_{j}^{∗}’s are lifts of elements of the UP G subgroup H_{0}|π_{1}(G_{i−1}) and that, since s_{i} is an action and the g_{j}’s commute, the ψ̂_{j}^{∗}’s commute. Lemma 3.4 therefore produces P_{i}, Q_{i} ∈ C_{∞}^{∗} that are fixed by each ŝ_{i}(g_{j}) and hence by ŝ_{i}(f) for each f ∈ K. The line L̃_{i} connecting P_{i} to Q_{i} is s_{i}(f)_{#}-invariant for each f ∈ K.

If L̃_{i} has the same image in G as L̃_{j} for some j < i, then after replacing Ẽ_{i}^{∗} by a translate if necessary, we may assume that L̃_{i} = L̃_{j}. Choose a highest edge splitting vertex w̃_{i} of L̃_{i} and apply the sliding operation of [BFH96a] simultaneously to each f ∈ K. This does not change s_{i}(f)|Γ_{i−1} and respects the group structure of K. A new Kolchin representative (still called K) is produced that agrees with the old one on G_{i−1} and has the additional feature that w̃_{i} is the terminal vertex of Ẽ_{i}^{∗}; in particular, the first condition of property A is satisfied. Since w̃_{i} can be any highest edge splitting vertex, we may assume without loss that the second condition of property A is also satisfied.

Suppose now that the kernel K of R is non-trivial. Each f^{∗} ∈ K satisfies E_{j} ↦ E_{j} for 1 ≤ j ≤ i−1 and E_{i} ↦ E_{i}u_{i,f^{∗}}. Thus the restriction of s_{i}(f^{∗}) to Γ_{i−1} is the covering translation determined by the lift of u_{i,f^{∗}} beginning at ṽ_{i}. The assignment f^{∗} ↦ u_{i,f^{∗}} defines a ‘suffix’ homomorphism from K into the free group π_{1}(G, v_{i}). Since K, and hence K, is solvable, the image of this homomorphism is isomorphic to Z. Thus the non-trivial s_{i}(f^{∗})|Γ_{i−1}’s have a common axis L̃_{i}. Choose f^{∗} so that s_{i}(f^{∗})|Γ_{i−1} is non-trivial. For each f ∈ K, s_{i}(ff^{∗}f^{−1})|Γ_{i−1} is a non-trivial covering translation with axis s_{i}(f)_{#}(L̃_{i}). Since K is normal in K, s_{i}(f)_{#}(L̃_{i}) = L̃_{i}. We have verified that L̃_{i} is s_{i}(f)_{#}-invariant for each f ∈ K. The proof now concludes as in the previous case.

4 Abelian subgroups are finitely generated

In this section we prove that an abelian UPG subgroup H_{0} is finitely generated. Section 3 and the exact sequence 1 → H_{0} → H →^{Ω} ℤ^{b} → 1 then implies that every solvable subgroup of Out(F_{n}) is finitely generated. This fact was originally proved by H. Bass and A. Lubotzky [BL94]. We also show that the above sequence is a virtually central extension.

If K is a Kolchin representative of H_{0}, f ∈ K and Φ_{K} : K → ℤ^{r} is the embedding of Lemma 3.3, then the coordinates of Φ_{K}(f) are defined in terms of the s_{i}(f)’s and the s_{α}(f)’s. Our goal is to recognize these lifts of f by their induced actions on C_{∞}, thus removing their dependence on the choice of K.

We begin placing an additional restriction (the second item below) on our Kolchin graphs and justifying our assumption from section 3 (the first item below).

Lemma 4.1. For every finitely generated UPG subgroup H_{0} there is a Kolchin representative K with the following properties.

• If E_{i} is not a loop then u_{i,f} is non- trivial for some f ∈ K.

• Every vertex of G is the initial vertex of at least two edges.

Proof of Lemma 4.1 Start with any Kolchin representative K. After restricting to a subgraph if necessary, we may assume that G has no valence one vertices. We will show that if either of the two properties fail, then we can replace G by a graph with fewer edges. This process terminates after finitely many steps to produce the desired Kolchin representative.

If the first property fails, then E_{i} is pointwise fixed for each f ∈ K and we may collapse it to a point.

Suppose then that the first property holds, that v is a vertex and that E_{i} is the first edge that contains v. If E_{i} is not a loop, then u_{i,f} is a non-trivial loop in G_{i−1} containing the terminal endpoint of E_{i} for some f ∈ K. Since loops are assumed to be immersed, they cannot pass through valence one vertices and v must be the initial vertex of E_{i}. For the same reason, v must also be the initial vertex of the second edge that is attached to it.

Suppose then that E_{i} is a loop and that E_{j_{0}}, …, E_{j_{m}} are the other edges that contain v. If the second property fails then E_{j_{0}}, …, E_{j_{m}} are non-loops with v as terminal endpoint and u_{j}(f) = E_{i}^{k_{j}(f)} for j = j_{0}, …, j_{m}. Redefine K by replacing each k_{j}(f) with k_{j}(f) − k_{j_{0}}(f). This can be achieved on the graph level by sliding v around E_{i} j_{0}(f) times. This has no effect on the outer automorphism determined by f and respects the group structure on K. The edge E_{j_{0}} is now fixed by each f ∈ K and so can be collapsed to point.

Definition 4.2. We say that a Kolchin representative K for H_{0} is an abelian Kolchin representative if it satisfies property A and the conclusions of Lemma 4.1.

We now turn to the task of determining, from the action of K on C_{∞} by lifts, if the axis of a given covering translation projects to an element of A(K).

Recall that in the analogy with the mapping class group of a compact surface M, A(K) corresponds to the set of reducing curves in the minimal reduction. Such reducing curves are completely characterized as follows (See Example 3.2, the proof of Proposition 1.2 or [HT85]): The free homotopy class determined by a closed curve α ⊂ M is an element of the minimal reducing set for the mapping class represented by the homeomorphism h : M → M if and only if for some (and hence each) covering translation T : M̃ → M̃ corresponding to α, there are two lifts h̃_{1}, h̃_{2} : M̃ → M̃ that commute with T and whose extensions ĥ_{i} over the ‘circle at infinity’ fix at least three points. (If the free homotopy class of α is fixed by h but α is not one of the reducing curves then there is one such lift h̃.) The analgous result in the non-geometric case is given in Corollary 4.6.

Definition 4.3. For any ψ ∈ H_{0} and covering translation T, define IL(ψ, T) (for Interesting Lifts) to be the set of lifts ψ̂ : C_{∞} → C_{∞} that commute with T and fix at least three points.

Recall that α ∈ A(K) has a preferred lift α̃ and an indivisible covering translation T_{α} with axis equal to α̃. The next lemma states that the actions s_{α} and s_{i} produce interesting lifts.

Lemma 4.4. Suppose that K is an abelian Kolchin representative and that α is the essential axis for E_{i} ∈ E(K). Then ⋂_{f∈K} F ix(ŝ_{α}(f)) and ⋂_{f∈K} F ix(ŝ_{i}(f)) each contain at least three points. In particular, if f represents ψ ∈ H_{0}, then ŝ_{α}(f), ŝ_{i}(f) ∈ IL(ψ, T_{α}).

Proof of Lemma 4.4 There is a preferred topmost splitting vertex ṽ_{α} ∈ α̃. Choose an edge Ẽ_{j} with initial vertex ṽ_{α}; by Lemma 4.1, we may assume that E_{j} ≠ α. If u_{j,f} is trivial for all f ∈ K, then E_{j} is a loop that is fixed by each f ∈ K. In this case, let R_{α} be an endpoint of the axis for E_{j} that contains Ẽ_{j}. If some u_{j,f} is non-trivial, then L̃_{j} is defined and we choose R_{α} to be an endpoint of the translate L̃′_{j} of L̃_{j} associated to Ẽ_{j}. In either case, each ŝ_{α}(f) fixes R_{α}.

The proof for s_{i}(f) is similar. By Lemma 4.1 there exists an edge Ẽ_{l} ≠ Ẽ_{i}^{∗}, with the same initial endpoint as Ẽ_{i}^{∗}. Define R_{i} ∈ Fix(ŝ_{i}(f)) as in the previous case using E_{l} in place of E_{j}.

If f ∈ K represents ψ ∈ H_{0}, then we use ŝ(f) and ŝ(ψ) interchangably. We refer to the ŝ_{i}(ψ)’s and the ŝ_{α}(ψ)’s as the canonical lifts of ψ. The next lemma and corollary show that A(K) depends only on H_{0} and not on the choice of K and that one can decide if ψ̂ is canonical from its action on C_{∞}.

Lemma 4.5. Suppose that K is an abelian Kolchin representative and that L̃ is a line with endpoints P, Q ∈ C_{∞}. Suppose further that:

• f̃: Γ → Γ is a lift of some f ∈ K

• Fix(f̂) contains P, Q and at least one other point.

• f̃ does not fix the highest edge splitting vertices of L̃ .

Then there exists i and a covering translation T such that T (L̃) = L̃_{i} and f̃ = T^{−1}s_{i}(f)T.

Proof of Lemma 4.5 By Lemma 2.2, Fix(f̃) ≠ ∅. Choose an arc σ̃ that intersects Fix(f̃) only in its initial vertex, say p̃_{1}, and that intersects the splitting vertices of L̃ only in its terminal vertex, say ṽ. Let Ẽ′_{i} be the first edge of σ̃ and let p̃_{2} be its terminal endpoint. Since f̃ fixes p̃_{1} but not p̃_{2}, u_{i,f} is non-trivial and L̃_{i} is defined. Let T be the covering translation that carries Ẽ_{i}^{∗} to Ẽ′_{i} and let L̃′_{i} = T (L̃_{i}) be the translate of L̃_{i} associated to Ẽ′_{i}. By construction, the ends of L̃′_{i} are f̂- invariant.

If L̃′_{i} ≠ L̃ then there is an endpoint, say S, of L̃′_{i} that is neither P nor Q. Lemma 2.2 implies that the highest edge splitting vertices of either L̃_{P S} (= the line connecting P to S) or L̃_{SQ} are fixed. But L̃_{P S} consists of a segment of L̃, a segment of L̃_{i} and perhaps a segment of (σ̃ \ Ẽ_{i}). The last segment contains no fixed vertices. By construction, the highest edge splitting vertices of the other segments are not fixed. Thus the highest edge splitting vertices of L̃_{P S} are not fixed. The symmetric argument applies to L̃_{SQ} and yields the desired contradiction. We conclude that L̃′_{i} = L̃ and that T (ṽ_{i}) = p̃_{2} = ṽ. Since T^{−1}s_{i}(f)T and f̃ both fix p̃_{1}, they must be equal.

Corollary 4.6. If K is an abelian Kolchin representative of H_{0}, then:

• A covering translation T corresponds to an essential axis of K if and only if IL(ψ, T) contains at least two elements for some ψ ∈ H_{0}.

• For each α ∈ A(K) and each ψ ∈ H_{0}, IL(ψ, T_{α}) = {ŝ_{i}(ψ) : E_{i} is an essential edge with axis α} ∪ {ŝ_{α}(ψ)}

• A(K) = A(H_{0}) depends only on H_{0} and not on K.

Proof of Corollary 4.6 The first and second items are a direct consequence of Lemma 4.4 and Lemma 4.5. The third item follows from the first.

Lemma 4.7. Every abelian UPG subgroup H_{0} is finitely generated.

Proof of Lemma 4.7 Choose an increasing sequence H_{1} ⊂ H_{2} ⊂ … of finitely generated subgroups whose union is H_{0}. The cardinality of A(H_{j}) and the multiplicities of the elements of A(H_{j}) are uniformly bounded (by the maximum number of edges in a marked graph with the property that the terminal vertex of each edge has valence at least three.) Since A(H_{j}) ⊂ A(H_{j+1}), we may assume after passing to a subsequence, that A(H_{j}) and the multiplicities are independent of j. In particular, there is a fixed r so that Φ_{j} : H_{j} → ℤ^{r} where Φ_{j} is the embedding of Lemma 3.3 and where H_{j} has been identified with a Kolchin representative K_{j}.

Fix ψ ∈ H_{1}, α ∈ A(H_{1}) and j ≥ 1; Let IL(ψ, T_{α}) = {b_{1}, b_{2}, … }. For each b_{k} and b_{l}, there is an integer p(k, l) such that b_{l}b_{k}^{−1} = T_{α}^{p(k,l)}. For any fixed f ∈ K, Corollary 4.6 implies that each coordinate ϕ_{j}^{i}(f) of Φ_{j}(f) is one of the p(k, l)’s. In particular, Φ_{j}(f) takes on only finitely many values as j varies.

Let g_{1}, …, g_{q} be generators of H_{1}. After passing to a subsequence we may assume that Φ_{j}(g_{i}) is independent of j for i = 1 … q. The lattice Φ_{j}(H_{1}) is therefore independent of j. It is contained with finite index in a maximal lattice L of rank q. The lattice Φ_{j}(H_{j}) has rank q and contains Φ_{j}(H_{1}) so is contained in L. In particular, the index of H_{1} in H_{j} is uniformly bounded. It follows that H_{j} = H_{j+1} for all sufficiently large j.

We close this section by showing that 1 → H_{0} → H →^{Ω} ℤ^{b} → 1 is a virtually central extension; the proof is a variation on that of Lemma 4.7.

Lemma 4.8. A(H_{0}) is H invariant (up to conjugacy). There is a finite index subgroup H′ ⊂ H whose actions on H_{0} and on A(H_{0}) are trivial.

Proof of Lemma 4.8 For each ψ ∈ H_{0}, each α ∈ A(H_{0}) and each lift η̂ of each η ∈ H, conjugation by η̂ sends IL(ψ, T_{α}) to IL(ψ′, T′) where ψ′ = ηψη^{−1} and T′ = η̂T_{α}η̂^{−1}. In particular, IL(ψ, T_{α}) and IL(ψ′, T′) have same cardinality. Corollary 4.6 therefore implies that the axis of T′ projects to an element of A(H_{0}) and so A(H_{0}) is invariant under the action of η. Since A(H_{0}) is finite, there is a finite index subgroup for which this action is trivial.

We assume now that the action on A(H_{0}) is trivial. Choose η̂ so that T′ = T_{α} or equivalently so that η̂ commutes with T_{α}. Choose an abelian Kolchin representative K for H_{0} and let Φ : H_{0} → ℤ^{r} be the embedding of Lemma 3.3. For any pair of elements b_{k}, b_{l} ∈ IL(ψ, T_{α}), there is an integer p(k, l) such that b_{l}b_{k}^{−1} = T_{α}^{p(k,l)}. Let P (ψ) be the finite collection of integers that occur as p(k, l)’s. Since conjugation by η̂ carries IL(ψ, T_{α}) to IL(ψ′, T_{α}) and since η̂ commutes with T_{α}, P (ψ) = P (ψ′). By construction, Φ(ψ′) therefore takes on only finitely many values as η varies over H and ψ ∈ H_{0} is fixed. Since Φ is an embedding, ψ′ takes on only finitely many values. After passing to a finite index subgroup we may assume that the action of H by conjugation on ψ is trivial. After applying this argument to a finite generating set for H_{0}, we see that the action of H on H_{0} is virtually trivial.

5 Proof of Proposition 1.2

The following lemma produces interesting lifts for (iterates of) individual elements of Out(F_{n}).

Lemma 5.1. Suppose that n ≥ 2 and that η ∈ Out(F_{n}). After replacing η by an iterate if necesssary, there is a lift η̂ ∈ EH(C_{∞}) that fixes at least three points. If γ is a non-trivial primitive element of F_{n} that is fixed (up to conjugacy) by η and if T is a covering translation corresponding to γ, then we may choose η̂ to commute with T.

Proof of Lemma 5.1 We assume at first that γ and T are given.

The case that η is realized as an isotopy class of a surface homeomorphism h : S → S is well known (see for example Lemma 3.1 of [HT85]): The Thurston classification theorem implies, after replacing h by an iterate if necessary, that S divides along annuli into subsurfaces with negative Euler characteristic on which h is either the identity or is pseudo-Anosov. Assuming that the reduction is done along the minimal number of annuli, we may choose the curve representing γ to lie in one of the subsurfaces S_{i}. If h|S_{i} is the identity, then it has a lift that fixes the ends determined by S_{i} (See Example 3.2). If h|S_{i} is pseudo-Anosov then γ determines a boundary component of S_{i} and there is a lift fixing γ̃ and the endpoints of the singular leaves of the pseudo-Anosov foliations associated to that boundary component.

We now turn to the general case and argue by induction. Since every outer automorphism of F_{2} is realized as a surface isotopy class, the preceding argument handles the n = 2 case. We may therefore assume that the lemma holds for free groups of rank less than n.

The smallest free factor F(γ) that contains γ [BFH96a] is η-invariant (up to conjugacy). If 1 < rank(F(γ)) < n, then the inductive hypothesis provides a lift of η|F(γ) with the desired properties. Extending this lift (Lemma 2.1) to all of F_{n} completes the proof. We may therefore assume that rank(F(γ)) is either 1 or n.

Choose an improved relative train track map f : G → G representing an iterate of η. Theorem 6.4 of [BFH96a] contains a list of all the properties of f : G → G that are used in this proof. The conjugacy class of γ determines a loop in G that we also call γ.

If F(γ) has rank n, then γ must cross an edge in the highest stratum of G. If this stratum is exponentially growing, then Theorem 6.4 of [BFH96a] implies that there is an η-invariant (up to conjugacy) subgroup 𝔽 that is its own normalizer and with the following additional property: There is a conjugacy between the the outer automorphism η|𝔽 and a pseudo-Anosov mapping class h : S → S. Moreover, the conjugacy carries γ to a boundary component of S. Since 𝔽 has rank at least two, lifts of η|𝔽 extend uniquely to lifts of η. We may therefore assume that 𝔽 = F_{n} and hence that η is represented by h. We are now reduced to a previous case.

If the top stratum G_{m} is not exponentially growing, then G_{m} is a single edge E_{m}. The loop γ splits (Lemma 5.2 of [BFH96a]) at the initial vertex v of E_{m} each time that it crosses E_{m} in either direction. We may therefore think of γ = γ_{1} · · · · · γ_{r} as a concatenation of Nielsen paths based at v (i.e. each [f(γ_{i})] = γ_{i}). We claim that there are two distinct γ_{l}’s. If not, then, since γ is indivisible, γ = γ_{1} is of the form E_{m}δ, δĒ_{m} or E_{m}δĒ_{m} for some path δ ⊂ G_{m−1}, where Ē_{m} is E_{m} with its orientation reversed. In the first two cases γ is a free factor and in the last case γ is freely homotopic to δ. Each of these contradicts our assumption that F(γ) has rank n and so verify our claim.

The axis Ax(T) = … γ̃_{1}γ̃_{2} … of T decomposes as a concatenation of lifts of the γ_{l}’s. Let ṽ be the initial vertex of γ̃_{1} and let f̃ be the lift of f that fixes ṽ. Since the γ_{l}’s are Nielsen paths, f̃ fixes each concatenation point in the decomposition Ax(T) = … γ̃_{1}γ̃_{2} . . .. In particular, η̂ = f̂ fixes the endpoints of Ax(T) and so commutes with T. Now think of γ_{1} as a loop and extend γ̃_{1} to the axis Ax(T_{1}) of a covering translation by concatenating translates of γ̃_{1}. Then η̂ = f̂ also fixes the endpoints of Ax(T_{1}).

It remains to consider the case that F(γ) has rank one. We may assume that the first stratum G_{1} = γ is a single edge; let v be the vertex of G_{1}.

Suppose that an edge of an exponentially growing stratum H_{r} is attached to v. After replacing f by an iterate if necessary there is an edge E of H_{r} with initial vertex E such that f(E) = E · β splits into the concatenation of E with some non-trivial path β. Let ṽ be a lift of v in the axis of T, let f̃: Γ → Γ be the lift of f that fixes ṽ and let Ẽ be the lift of E with initial vertex ṽ. Then f̃(Ẽ) = Ẽ · β̃ splits into the concatenation of Ẽ and some other non-trivial path and so each [f̃^{k}(Ẽ)] is a proper initial segment of [f̃^{k+1}(Ẽ)]. It follows that the [f̃^{k}(Ẽ)]’s converge to an invariant ray whose endpoint R is fixed by f̃. Let η̂ = f̂.

If there are no exponentially growing strata attached to v, then there are no zero strata attached to v and each edge E_{i} that is attached to v is its own stratum H_{i} and satisfies f(E_{i}) = E_{i}u_{i} for some path u_{i} ⊂ G_{i−1}. If v is the initial vertex of E_{i} and either E_{i} is a loop or u_{i} is non-trivial, then we define η̂ as in the proof of Lemma 4.4: Let ṽ be a lift of v in the axis of T, let f̃: Γ → Γ be the lift of f that fixes ṽ and let Ẽ be the lift of E with initial vertex ṽ. If u_{i} is trivial then R is the endpoint of the axis for E_{i} that contains Ẽ_{i}. If u_{i} is non-trivial then R is the endpoint of the invariant ray ũ_{i} · [f̃(ũ_{i})] · [f̃^{2}(ũ_{i})] · . . ..

In the remaining case, every E_{i} attached to v is either a fixed non-loop or has v as its terminal endpoint. Proceeding as in the proof of Lemma 4.1 we can reduce the number of edges in G and arrive at one of our previous cases.

This completes the proof when γ and T are given. It remains to consider the case that η does not act periodically on any conjugacy class in F_{n}. By induction on n, we may assume that η does not act periodically on the conjugacy class of any proper free factor in F_{n}. We may therefore assume our improved relative train track map f : G → G has only one stratum and that this stratum is exponentially growing.

After passing to a further iterate if necessary, we may assume that there is a vertex v and two edges E_{1} and E_{2} initiating at v such that f(E_{i}) = E_{i} · β_{i} splits into the concatenation of E_{i} with some non-trivial path β_{i}. Choose a lift ṽ of v, let f̃ be the lift of f that fixes ṽ and let Ẽ_{i} be the lift of E_{i} initiating at ṽ. As above, [f̃^{k}(Ẽ_{i})] converges to an invariant ray whose endpoint R_{i} is fixed by f̂. Moreover, the bounded cancellation lemma [Coo87] and the fact that the lengths of [f̃^{k}(Ẽ_{i})] grow exponentially in k imply that R_{i} is an attracting fixed point for the action of f̂ on C_{∞}.

We have shown that some iterate of η has a lift with at least two attracting fixed points. Applying this to η^{−1}, we conclude (suppressing the iterate for notational simplicity) that some η̂ has at least two repelling fixed points. If η̃ has a fixed point, then the preceding argument shows,after passing to an iterate if necessary, that there are also at least two attracting fixed points. Suppose then that η̃ is fixed point free. During the proof of Proposition 6.21 of [BFH96a] we show that there exists x̃ ∈ Γ such that x̃, η̃(x̃), η̃^{2}(x̃) … is an infinite sequence in an embedded ray B̃; the endpoint of B̃ is fixed by η̂ and is not one of the repelling fixed points.

We prove Proposition 1.2 in the following equivalent form.

Proposition 1.2 Every abelian subgroup H ⊂ Out(F_{n}) has a virtual lift Ĥ ⊂ EH(F_{n}). If γ is a non-trivial primitive element of F_{n} that is fixed, up to conjugacy, by each element of H and if T is a covering translation corresponding to γ, then Ĥ can be chosen so that each element commutes with T.

Proof of Proposition 1.2 We argue by induction on the rank of the free abelian group H. If H has rank one with generator η, then Ĥ is determined by choosing a lift η̂ that commutes with T if T is given. We may now assume that the rank of H is at least two and that the lemma holds for all ranks less than that of H.

By Lemma 5.1, there is an element η ∈ H and a lift η̂ that fixes at least three points and that commutes with T if T is given. Let C = F ix(η̂) and let T (C) be the group of covering translations that preserve C. As described in subsection 2.1, η̂ determines an automorphism Φ : F_{n} → F_{n} whose fixed subgroup 𝔽 corresponds to T (C) under the identification of F_{n} with T . (See the proof of Lemma 6.2 for further details.) If γ and T are given, then T ∈ T (C) and γ ∈ 𝔽.

We first show, after passing to a subgroup of H with finite index, that every μ ∈ H has a lift μ̂ that commutes with η̂ and with T if T is given. In particular, C is μ̂-invariant, 𝔽 is H-invariant (up to conjugacy) and, if γ is given then each element of H|𝔽 fixes γ up to conjugacy.

Suppose at first that γ and T are not given. We say that two lifts η̂_{1} and η̂_{2} of η are equivalent if η̂_{1} = T_{1}η̂_{2}T_{1}^{−1} for some covering translation T_{1}. Let IL(η) be the set of equivalence classes of lifts η̂ that fix at least three points. By Lemma 2.3, IL(η) is finite. Since H is abelian, H acts by conjugation on IL(η). After passing to a subgroup of finite index, we may assume that this action is trivial. Thus for any lift μ̂_{1} of μ ∈ H, there is a covering translation T_{1} such that μ̂_{1}η̂μ̂_{1}^{−1} = T_{1}η̂T_{1}^{−1}. Thus μ̂ = T_{1}^{−1}μ̂_{1} commutes with η̂.

Suppose now that γ and T are given. Each μ ∈ H has a lift μ̂ that commutes with T. Since H is abelian, μ̂η̂μ̂^{−1} is a lift of η that commute with T and so μ̂η̂μ̂^{−1} = T^{a}η̂ for some a. Let η̂_{k} = μ̂^{k}η̂μ̂^{−k} = μ̂η̂_{k−1}μ̂^{−1} = T^{ak}η̂. Then each η̂_{k} is conjugate to η̂ and so fixes at least three points. On the other hand, there are only finitely many values of l for which F ix(T^{l}η̂) ≠ F ix(T). (This follows from: (i) η̂ fixes F ix(T) = {P, Q} and so cannot move points very near P to points very near Q; and (ii) T acts cocompactly on C_{∞} \ {P, Q}.) We conclude that a = 0 and hence that μ̂ commutes with η̂.

The proof now divides into cases, depending on the rank of T (C). If T (C) is the trivial group, then each μ has a unique lift μ̂ that preserves C and so the assignment μ ↦ μ̂ defines Ĥ.

Suppose next that T (C) has rank one. Since 𝔽 is H-invariant (up to conjugacy), we may assume that T is given and generates T (C).

We claim that C contains only finitely many T -orbits. This is a special case of the main theorem of [Coo87]; the argument in this case is short so we include it for completeness. Let P and Q be the endpoints of the axis of T. Since T acts cocompactly on C_{∞} \{P, Q}, it suffices to show that P and Q are the only accumulation points of C. Suppose to the contrary that S is an accumulation point other than P and Q. Arguing as in the proof of Corollary 2.3, using triples of points in C limiting on S, we find lifts ṽ_{i} ∈ Γ of a vertex v ∈ G so that the covering translation T_{i} that carries ṽ_{1} to ṽ_{i} commutes with η̂. But then T_{i} is a multiple of T in contradictiction to the assumption that T_{i}(ṽ_{1}) → S. This verifies our claim.

The action of H on the finitely many T -orbits of C is well defined. After passing to a finite index subgroup, this action is trivial. Choose a point R ∈ C \ {P, Q}. Composing μ̂ with an iterate of T, we may assume that μ̂ fixes R. The assignment μ ↦ μ̂ defines Ĥ.

Finally suppose that T (C) has rank at least two. As noted above, if γ is given then it is fixed, up to conjugacy, by each element of H^{∗} = H|𝔽. Since 𝔽 is the fixed subgroup of an automorphism Φ representing η, the image of η in H^{∗} is trivial. Thus the rank of H^{∗} is less than that of H and by induction, there is a lift Ĥ^{∗} ⊂ EH(C) to elements that commute with T if T is given. By Lemma 2.1, Ĥ^{∗} extends to the desired lift Ĥ.

6 Proof of Theorem 1.1

We may assume without loss that H_{0} is non- trivial and, by Lemma 4.8, that H acts trivially on H_{0} and on A(H_{0}). Let K be an abelian Kolchin representative for H_{0}. Throughout this section ŝ = ŝ_{i} or ŝ_{α} and T_{α} is its associated covering translation.

Before proving the next lemma we show that it implies Theorem 1.1.

Lemma 6.1. The lift ŝ(H_{0}) ⊂ EH(C_{∞}) virtually extends to a lift Ŝ(H) ⊂ EH(C_{∞}) all of whose elements commute with T_{α}.

Proof of Theorem 1.1 We may assume that the extensions Ŝ_{i} and Ŝ_{α} of ŝ_{i} and ŝ_{α} produced by Lemma 6.1 are defined on all of H. Define a homomorphism ϕ′_{i} : H → ℤ by S_{i}(ψ) = T_{α}^{ϕ′_{i}(ψ)}S_{α}(ψ) and note that the product Φ′ : H → ℤ^{r} of the ϕ′_{i}’s extends the embedding Φ : H_{0} → ℤ^{r} of Lemma 3.3. Define Ψ : H → ℤ^{b+r} to be the product of Ω and Φ′. Since H_{0} is the kernel of Ω and Φ′|H_{0} = Φ is an embedding, Ψ is an embedding.

We now know that solvable subgroups H of Out(F_{n}) are finitely generated and virtually abelian. By passing to a subgroup of index at most D(n) where D(n) := |GL(n, ℤ/3ℤ)| < 3^{n^{2}}, we may assume that the image of H in GL(n, ℤ/3ℤ) is trivial. Thus, H has a subgroup of index at most this number that is a torsion free Bieberbach group of vcd at most vcd(Out(F_{n})) = 2n − 3 (see [CV86]). Also, a Bieberbach group of vcd at most n has a subgroup of index at most D(n) that is free abelian (see, for example, [Cha86]). Thus, H has a free abelian subgroup of index at most D(n)D(2n − 3) < 3^{5n^{2}}. This completes the proof of the Theorem 1.1.

We are now reduced to Lemma 6.1. The proof uses Proposition 1.2 and follows the general line of the proof of Proposition 1.2.

Let C = ⋂_{f∈K} F ix(ŝ(f)) and let T (C) be the group of covering translations that preserves C. Note that T (C) contains T_{α} and so has rank at least one. Let 𝔽 be the subgroup of F_{n} corresponding to T (C) and let s̃(H_{0}) ⊂ Aut(F_{n}) be the lift of H_{0} corresponding to ŝ(H_{0}) ⊂ EH(C_{∞}).

Lemma 6.2. 𝔽 is the fixed subgroup {γ ∈ F_{n} : η̃(γ) = γ for each η̃ ∈ H̃_{0}} of s̃(H_{0}).

Proof of Lemma 6.2 If the endpoints of the axis of T ∈ T are contained in C, then they are fixed by each ŝ(f) and so T commutes with each s(f) : Γ → Γ. Thus T commutes with each ŝ(f) and each F ix(ŝ(f)) is T -invariant. It follows that T ∈ T (C).

Conversely, if P and Q are, respectively, the backward and forward endpoints of the axis of T then lim_{n→∞} T^{n}(R) = Q and lim_{n→∞} T^{−n}(R) = P for all R ∈ C_{∞} \ {P, Q}. If T (C) = C then C must contain P and Q.

We have shown that T ∈ T (C) if and only if the endpoints of the axis of T are contained in C. By construction, the latter condition is equivalent to the endpoints of the axis of T being fixed by each ŝ(f). The lemma now follows from the definition of s̃(H).

Proof of Lemma 6.1 Given μ ∈ H, choose a lift μ̂ that commutes with T_{α}. Suppose also that η̂ ∈ ŝ(H_{0}) is given. Since H acts trivially on H_{0}, μ̂η̂μ̂^{−1} is a lift of η that commute with T_{α} and so μ̂η̂μ̂^{−1} = T_{α}^{a}η̂ for some a. Arguing exactly as in the proof of Proposition 1.2, we conclude that μ̂ commutes with η̂. It follows that C is μ̂-invariant, that 𝔽 is H- invariant up to conjugacy and that each element of H|𝔽 fixes α up to conjugacy.

The proof now divides into cases, depending on the rank of T (C). Suppose that T (C) has rank one. We claim that C contains only finitely many T_{α}-orbits. Let P and Q be the endpoints of the axis of T_{α}. Since T_{α} acts co-compactly on C_{∞} \{P, Q}, it suffices to show that P and Q are the only accumulation points of C. Suppose to the contrary that S is an accumulation point other than P and Q. Lemma 2.2, applied to triples of points in C limiting on S, implies that there are vertices ṽ_{i} ∈ Γ that limit on S and that are fixed by s(f) for each f ∈ K. There is no loss in assuming that the v_{i}’s are all lifts of the same vertex in G. The covering translation T_{i} that carries ṽ_{1} to ṽ_{i} commutes with each s(f) and so must be a multiple of T_{α}. But this contradicts the assumption that T_{i}(ṽ_{1}) → S. This verifies our claim.

There is a well defined action of H on the finitely many T -orbits of C. After passing to a finite index subgroup, this action is trivial. By Lemma 4.4 there exists R ∈ C_{∞} that is not an endpoint of the axis of T_{α}. Composing μ̂ with an iterate of T, we may assume that μ̂ fixes R. The assignment η ↦ η̂ defines Ŝ.

We may now assume that T (C) has rank at least two. By Lemma 2.1, the restriction H^{∗} of H to Out( 𝔽) is well defined. By Lemma 6.2, the restriction H_{0}^{∗} of H_{0} to Out(𝔽^{∗}) is trivial. Restricting the exact sequence 1 → H_{0} → H →^{Ω} ℤ^{b} → 1 to 𝔽^{∗} we see that H^{∗} is abelian. By Propositon 1.2, there is a virtual lift Ŝ^{∗} ⊂ EH(C) of H^{∗} such that each element of Ŝ^{∗} commutes with T_{α}. Let Ŝ be the unique extension of Ŝ^{∗} to a virtual lift of H. Since H_{0}^{∗} is trivial, Ŝ(ψ) restricts to the identity on C for each ψ ∈ H_{0}. Thus Ŝ(ψ) and ŝ(ψ) agree on C and so must agree everywhere.

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