Direct Products of Free Groups in Aut(F_N)

Abstract

We give a complete description of the embeddings of direct products of nonabelian free groups into Aut(F_N) and Out(F_N) when the number of direct factors is maximal. To achieve this, we prove that the image of each such embedding has a canonical fixed point of a particular type in the boundary of Outer space.

1 Introduction

Mapping class groups of surfaces of finite type and automorphism groups of free groups are central objects in geometric topology and group theory, and it is natural to study them in parallel. Nielsen-Thurston theory [Thu88] provides a potent geometric description of the individual elements of mapping class groups, and the train-track technology initiated by Bestvina and Handel [BH92] provides an equally potent description in the wilder setting of free group automorphisms. In pursuit of a more global understanding of these groups, one seeks insight from their actions on Teichmüller space and Outer space, as well as associated spaces such as the curve complex, in the case of mapping class groups, and free factor and splitting complexes in the case of Aut(F_N) and Out(F_N).

The need for geometric insights and invariants comes into particularly sharp focus when one is trying to elucidate the intricate subgroup structure of these groups, as we are in this article. One sees this clearly in the classification of abelian subgroups, which has many ramifications. From the Nielsen-Thurston theory, one knows that, up to finite-index, every abelian subgroup of the mapping class group is generated by combinations of Dehn twists in a collection of disjoint curves and, optionally, a pseudo-Anosov automorphism on each connected component of the complement of these curves [BLM83, Iva92]. This description provides a host of geometric invariants for studying the totality of abelian subgroups, starting with the stable laminations of the pseudo-Anosov pieces and compatibility conditions for the curve systems. With these in hand, one can organise the commensurability classes of maximal-rank abelian subgroups into a space that is closely related to the curve complex. This idea is central to Ivanov’s proof [Iva97] of commensurator rigidity for mapping class groups: he proved that, with some low-genus exceptions, every isomorphism between finite-index subgroups of a mapping class group is the restriction of a conjugation in the ambient group. Although the situation in free groups is much more complicated, Feighn and Handel [FH11] succeeded in describing all abelian subgroups of Out(F_N), and this description was used by Farb and Handel [FH07] to establish commensurator rigidity for Out(F_N) in the case N≥4.

Ivanov’s commensurator rigidity theorem was later extended by Bridson, Pettet and Souto [BPS11] to various subgroups of the mapping class group; they followed a similar template of proof but used direct products of nonabelian free groups in place of abelian subgroups, replacing the curve complex with a complex built from decompositions of the surface into subsurfaces of Euler characteristic −2 (cf. [BM19]). In the same spirit, by focussing on direct products of nonabelian free groups rather than abelian subgroups, Horbez and Wade [HW20] proved that Out(F_N) has commensurator rigidity for N≥3, as do many of its natural subgroups.

Our main purpose in this article is to give a complete classification of the maximal-rank direct products of free groups in Aut(F_N) and Out(F_N); we shall see that they are remarkably rigid. More generally, we will work with maximal commuting families of full-sized groups (groups that contain non-abelian free subgroups). In a companion to this paper [BW24], we shall use this classification, in harness with [BB23], to prove that Aut(F_N) and its Torelli subgroup are commensurator rigid if N≥3.

The most important step in our proof of the classification is a fixed-point theorem that we establish for the action of Out(F_N) on the space of free splittings of F_N (Theorem A). To motivate this theorem, we begin by describing an example of a subgroup of Out(F_N) that is a direct product of the maximal number of copies of F₂; Horbez and Wade [HW20] proved that this number is 2N−4 (one less than the cohomological dimension).

We fix a basis {a₁,a₂,x₁,…,x_N−2} of F_N and consider the direct product D of the 2N−4 copies of F₂ in Out(F_N) obtained by multiplying the elements x₁,…,x_N−2 on the left and right by elements of 〈a₁,a₂〉. This group D fixes a graph-of-groups decomposition of F_N with a single vertex group given by A=〈a₁,a₂〉 and N−2 loops with trivial edge stabilizers (with x₁,…,x_N−2 as the stable letters). The Bass–Serre tree associated to any such decomposition determines an open simplex in the boundary of Culler and Vogtmann’s Outer space [CV86]; we call such a graph of groups a collapsed rose with N−2 petals (see Fig. 1).

Figure 1

A collapsed rose with N−2 petals.

Our first theorem shows that if a direct product of nonabelian free groups in Out(F_N) has the maximal number of factors, then it has a canonical fixed point of this type. Note that in the following theorem, and throughout, we do not assume that D is finitely generated.

Theorem A

Let N≥3 and suppose D<Out(F_N) is generated by a commuting family of 2N−4 full-sized subgroups. Then there is a unique collapsed rose with N−2 petals that is fixed by D.

When a subgroup of Out(F_N) fixes a tree T, the preimage of this group in Aut(F_N) admits an action on T. With a small amount of extra work, the following result can be deduced from Theorem A.

Theorem B

Let N≥3 and suppose D<Aut(F_N) is generated by a commuting family of 2N−3 full-sized subgroups. Then, the image of D in Out(F_N) fixes a unique collapsed rose with N−2 petals, and D acts on the Bass–Serre tree of this collapsed rose with a unique global fixed point.

In order to move from Theorems A and B to the precise algebraic description of the direct products of free groups that we seek, some more notation is required. We continue to work with a fixed basis {a₁,a₂,x₁,…,x_N−2} of F_N and let A be the free factor generated by a₁ and a₂. Let L_i be the free group of rank 2 in Aut(F_N) consisting of elements that send x_i↦ax_i for some a∈A and fix all other basis elements. Similarly, we use R_i to denote the free group of right transvections of x_i by elements of A. Furthermore, we let I(A) be the group of inner automorphisms generated by elements of A and let τ be the Nielsen automorphism mapping a₁↦a₁a₂ and fixing all other basis elements. We let $L_{i}^{\tau}$, $R_{i}^{\tau}$, and I(A)^τ be the respective subgroups of these groups that commute with τ (equivalently, the elements from A used in their associated transvections or inner automorphisms belong to ${\mathrm{{Fix}}}(\tau ) \cap A = \langle a_{1}a_{2}a_{1}^{-1}, a_{2} \rangle$).

Theorem C

Let N≥3 and suppose D<Aut(F_N) is a direct product of 2N−3 nonabelian free groups. Then a conjugate of D is contained in one of the following groups.

• L₁×⋯×L_N−2×R₁×⋯×R_N−2×I(A)

• $L_{1}^{\tau }\times \cdots \times L_{N-2}^{\tau }\times R_{1}^{\tau } \times \cdots \times R_{N-2}^{\tau }\times I(A)^{\tau }\times \langle \tau \rangle$

• $\langle L_{1}, \tau \rangle \times L_{2}^{\tau }\times \cdots \times L_{N-2}^{\tau }\times R_{1}^{\tau }\times \cdots \times R_{N-2}^{ \tau }\times I(A)^{\tau}$

• $L_{1}^{\tau }\times \cdots \times L_{N-2}^{\tau }\times R_{1}^{\tau } \times \cdots \times R_{N-2}^{\tau }\times \langle I(A), \tau \rangle$

In the second case, if D=D₁×⋯×D_2N−3, then each summand D_i is contained in a unique subgroup of the form $\langle L_{i}^{\tau}, \tau \rangle$, $\langle R_{i}^{\tau}, \tau \rangle$, or 〈I(A)^τ,τ〉. In the remaining cases, the inclusion respects the visible direct product decompositions, up to permuting the summands.

In Sect. 7 we shall prove that every direct product of 2N−4 nonabelian free groups in Out(F_N) is contained in the image of one of the subgroups listed in Theorem C. Furthermore, the enveloping groups are canonical in the following sense:

Theorem D

Let N≥3 and suppose D<Aut(F_N) is generated by a set of 2N−3 pairwise-commuting, full-sized subgroups. Then D is maximal with respect to inclusion among such subgroups of Aut(F_N) if and only if D is conjugate to one of the groups in the first, third, or fourth bullet points in Theorem C.

Theorem C places each summand D_i<D inside a group of the form F₂ or $F_{2} \times \mathbb{Z}$ or $M_{\tau}=F_{2}\rtimes _{\tau}\mathbb{Z}$. It is obvious that every free subgroup of F₂ or $F_{2} \times \mathbb{Z}$ is contained in a maximal free subgroup. In Sect. 8, we shall prove the less obvious fact that this is also the case in M_τ.

Corollary E

Let N≥3 and let $\mathcal{D}$ be the family of subgroups of either Aut(F_N) or Out(F_N) that are direct products of 2N−3 or 2N−4 nonabelian free groups, respectively. Then every $D \in \mathcal{D}$ is contained in a maximal element (with respect to inclusion).

A further consequence of Theorem C is that the centralizer of the direct product D is cyclic, and when it is non-trivial it is generated by a Nielsen automorphism. This yields the following rigidity result, which plays a crucial role in [BW24].

Theorem F

Let Γ be a finite-index subgroup of Aut(F_N) with N≥3 and let f:Γ→Aut(F_N) be an injective homomorphism. Every power of a Nielsen automorphism is mapped to a power of a Nielsen automorphism under f.

1.1 Techniques and proofs

In the remainder of this introduction, we shall describe the structure of this paper and sketch some of the main ideas that go into the proofs of the main results. Throughout, we assume N≥3. For ease of exposition, we restrict to the case where D is a direct product of free groups.

Our first goal is to establish the existence of the fixed point described in Theorem A. Let D be a direct product of 2N−4 nonabelian free groups in Out(F_N) and let $\widetilde{D}$ be its preimage in Aut(F_N). Our starting point is Theorem 6.1 from [HW20], where actions of Out(F_N) on relative free factor complexes were used to show that D fixes a one-edge nonseparating free splitting of F_N. Lemma 2.7 tells us that a simplicial tree T in the boundary of Outer space will be fixed by D if and only if the F_N-action on T extends to an action of $\widetilde{D}$ on T (where F_N is identified with the group of inner automorphisms in $\widetilde{D}$). We apply this to the one-edge splitting fixed by D, blowing-up the action of $\widetilde{D}$ on the Bass–Serre tree to obtain the action on a collapsed rose with N−2 petals that we seek; this blow-up, which is described in Sect. 6, is constructed using a graph of actions in the sense of Levitt [Lev94]. A key point is to argue that the stabilizer in $\widetilde{D}$ of a vertex v∈T acts on a collapsed rose with N−3 petals, and that the adjacent edge stabilizers $\widetilde{D}_{e}$ for the one-edge splitting are elliptic in this new action; this last property implies the edges of the old tree can be glued onto the new tree in a coherent fashion.

The uniqueness of the fixed point described in Theorem A is tackled separately. By work of Guirardel and Horbez [GH21, Sect. 6], if there were two collapsed roses with N−2 petals fixed by D then they would belong to the same deformation space, and a folding path between these two collapsed roses would have to be fixed by D. However, D is too large to fix any graph of groups decomposition of F_N with more than one vertex group, which means the folding path must be trivial. Details are given in Sect. 5.

Section 4 contains an analysis of the stabilizers in Aut(F_N) of collapsed roses and similar graphs. This analysis plays a significant role in the proof of Theorem A, and it renders the deduction of Theorem B straightforward, as we shall see at the end of Sect. 6. The analysis of stabilizers of collapsed roses also provides a crucial bridge from Theorem A to Theorem C. In particular, with Theorem A in hand, Proposition 4.1 essentially reduces Theorem C to an analysis of the ways in which a direct product of k+1 nonabelian free groups can embed in

$$ M_{k}(F_{2}) := F_{2}^{k} \rtimes {\mathrm{{Aut}}}(F_{2}), $$

where the action of Aut(F₂) in this semidirect product is diagonal. These embeddings are described in Theorem 3.8; the required algebra is surprisingly delicate. Given that our main results involve free groups of higher rank, it seems incongruous that special features of Aut(F₂) should play a crucial role at this stage of the proof, but nevertheless this is the case. A key fact that makes many arguments in Sect. 3 work is that powers of Nielsen transformations are the only automorphisms of F₂ that have nonabelian fixed subgroups [CT96]. This special property lies behind the appearance of the Nielsen transformation τ in Theorem C.

2 Product rank, splittings, and automorphic lifts

2.1 Direct products of free groups and commuting families of full-sized groups

A group G has product rank rk_F(G)=k if k is the largest integer such that G contains a direct product of k nonabelian free groups (possibly rk_F(G)=∞). To understand how product rank behaves with respect to homomorphisms between groups, we make use of the following standard lemma.

Lemma 2.1

Let K be a normal subgroup of a direct product of nonabelian free groups G₁×G₂×⋯×G_k. Suppose that rk_F(K)=l. Then, after reordering the factors, K is a normal subgroup of G₁×G₂×⋯×G_l×1×⋯×1.

Proof

Let g=(g₁,g₂,…,g_k)∈K and without loss of generality assume g₁≠1. Let h∈G₁ be an element that does not commute with g₁. Conjugation by (h,1,1,…,1) shows that (hg₁h⁻¹,g₂,g₃,…,g_k)∈K, so $(hg_{1}h^{-1}g_{1}^{-1},1,1,\ldots ,1) \in K$. Hence if K has a nontrivial projection to a factor then it intersects that factor in an infinite normal (hence nonabelian) subgroup. As rk_F(K)=l, it intersects exactly l factors. □

The following easy consequence of Lemma 2.1 is a variation on [HW20, Lemma 6.3].

Lemma 2.2

If H is a finite-index subgroup of G then rk_F(H)=rk_F(G). If

$$ 1 \to N \to G \to Q \to 1 $$

is an exact sequence of groups then rk_F(G)≤rk_F(N)+rk_F(Q).

Recall that a group is full-sized if it contains a nonabelian free subgroup.

Proposition 2.3

Let G be a group generated by a set G₁,…,G_k of commuting full-sized subgroups. Then rk_F(G)≥k. Furthermore, suppose K is a normal subgroup of G with rk_F(K)=l<k. Then, after permuting the G_i, the groups G_l+1/K,…,G_k/K form a commuting family of full-sized subgroups in G/K.

Proof

As each G_i is full-sized, each G_i contains a group D_i≅F₂. Let

$$ \phi \colon D_{1} \times D_{2} \times \cdots \times D_{k} \to G $$

be the homomorphism induced by mapping each D_i into G. We claim that ϕ is injective. Indeed, if K=kerϕ is nontrivial then there exists i such that K has a nontrivial projection to D_i, and as in the proof of Lemma 2.1, the intersection K∩D_i is nontrivial. However, each D_i injects into G under ϕ, which is a contradiction. Hence ϕ is injective and rk_F(G)≥k. Now suppose that K is a normal subgroup of G with rk_F(K)≤l. Let D=〈D₁,…,D_k〉, which is isomorphic to D₁×D₂×⋯×D_k by the work above. Let K′=K∩D. Then by Lemma 2.2, after reordering the G_i the group K′ is a normal subgroup of D₁×D₂×⋯×D_l×1×⋯×1, and D_i embeds in G_i/K for i=l+1,…,k. Hence these quotient groups are also full-sized. □

2.2 The product rank of Aut(F _N) and Out(F _N)

Fixing a basis $\mathcal{B}= \{a_{1},a_{2}, x_{1},\ldots ,x_{N-2}\}$ for F_N, one obtains a direct product of 2N−4 free groups of rank 2 in Aut(F_N) as follows. For i=1,…,N−2 let L_i be the subgroup consisting of automorphisms of the form [x_i↦wx_i, x_j↦x_j (j≠i)], where w is a word in the free group on {a₁,a₂}, and let R_i be the subgroup consisting of automorphisms of the form [x_i↦x_iw, x_j↦x_j (j≠i)]. Each L_i and R_i is a free group of rank 2, and these subgroups generate a direct product $D_{\mathcal{B}}= L_{1}\times R_{1}\times \cdots \times L_{N-2}\times R_{N-2}< {\mathrm{{Aut}}}(F_{N})$. As $D_{\mathcal{B}}$ contains no inner automorphisms, it injects into Out(F_N). Theorem 6.1 of [HW20] shows that Out(F_N) does not contain a direct product of 2N−3 nonabelian free groups if N>2, thus

$$ {{\mathrm{rk}_{F}}}({\mathrm{{Out}}}(F_{N}))=2N-4 $$

for N≥3. (Note that the virtual cohomological dimension of Out(F_N), which gives an upper bound on product rank, is 2N−3.)

The conjugations of F_N by a₁ and a₂ generate a further free subgroup I(A)<Aut(F_N) that commutes with $D_{\mathcal{B}}$. As $I(A)\cap D_{\mathcal{B}}$ is trivial, we get

$$ {{\mathrm{rk}_{F}}}({\mathrm{{Aut}}}(F_{N})) \ge 2N-3, $$

and by Lemma 2.2 we must have equality when N≥3. Since Aut(F₂) does not contain a direct product of two nonabelian free groups [Gor04] (see also Corollary 3.4 (4) below), we have equality in the case N=2 as well.

We summarize this discussion for later use:

Proposition 2.4

[HW20], Theorem 6.1

For every N≥2 we have

$$ {{\mathrm{rk}_{F}}}({\mathrm{{Aut}}}(F_{N})) =2N-3. $$

For every N≥3 we have

$$ {{\mathrm{rk}_{F}}}({\mathrm{{Out}}}(F_{N}))=2N-4. $$

Since ${\mathrm{{Out}}}(F_{2})\cong {\mathrm{{GL}}}(2,\mathbb{Z})$ is virtually free, rk_F(Out(F₂))=1.

2.3 Splittings and their stabilizers

A splitting of a group G is a minimal, simplicial left action on a tree. (The terminology comes from the fact that the quotient graph of groups splits G in terms of amalgamated free products and HNN extensions [Ser80].) The splitting is said to be free if all edge stabilizers are trivial. Two splittings T and T′ are deemed equivalent if there is a G-equivariant simplicial isomorphism from T to T′. The trees that we consider are not allowed to have vertices of valence two. (The quotient graph of groups may still have vertices v of valence two, in which case the vertex group G_v will be nontrivial.) We say that T′ is a collapse of T if the action of G on T′ is obtained by equivariantly collapsing a forest in T. Going in the opposite direction, we say that T is a refinement of T′ if T′ is a collapse of T. Two splittings are said to be compatible if they have a common refinement.

We shall be concerned almost entirely with the case G=F_N.

Each vertex stabilizer of a free splitting of F_N is a free factor. We work with the standard left action of Aut(F_N) on F_N. There is then a right action of Aut(F_N) on the set of all free splittings of F_N: the action of ϕ∈Aut(F_N) sends f:F_N→Isom(T) to f∘ϕ. This action respects equivalence classes of F_N-trees, and the inner automorphisms leave each equivalence class invariant. Thus there is an induced action of Out(F_N) on the set of equivalence classes of free splittings of F_N. Stabilizers under this action have been studied extensively in the literature; the most general results (replacing F_N with an arbitrary group and allowing more general splittings) appear in work of Bass–Jiang and Levitt [BJ96, Lev05].

We write ${\mathcal{FS}}$ for the set of equivalence classes of free splittings of F_N. When there is no danger of ambiguity, we shall not distinguish between a free splitting T and its equivalence class [T].

Unpacking the definitions, we see that $[T]\in {\mathcal{FS}}$ is fixed by an outer automorphism Φ∈Out(F_N) if and only if for each representative ϕ∈Φ there is a homeomorphism f_ϕ:T→T such that

$$ f_{\phi}(gx)=\phi (g)f_{\phi}(x) $$

(1)

for all x∈T and g∈F_N; in other words, [T] is fixed by ϕ∈Aut(F_N) if and only if there is an isomorphism from T to itself that is ‘ϕ-twistedly equivariant’. The map f_ϕ is unique. (This is true, more generally, for stabilizers of minimal irreducible G-trees.) If ϕ is conjugation by g, then f_ϕ(x)=gx.

We use Stab(T) to denote the stabilizer of [T] in Out(F_N). There is a homomorphism

$$ {\mathrm{Stab}}(T) \to {\mathrm{{Aut}}}(T/F_{N}) $$

given by the left action of each outer automorphism on the F_N-orbits of edges and vertices in T. We call the kernel of this map Stab⁰(T). (Here, T/F_N is the quotient graph, not the quotient graph of groups.)

We use $\mathrm{{\mathrm{IA}}_{N}(\mathbb{Z}}/\mathrm{3\mathbb{Z})}$ to denote the kernel of the map

$$ {\mathrm{{Out}}}(F_{N}) \to \mathrm{GL}_{N}(\mathbb{Z}/3\mathbb{Z}) $$

given by the action of Out(F_N) on $H_{1}(F_{N},\mathbb{Z}/3\mathbb{Z})$. The analogous subgroup of the mapping class group consists of pure mapping classes [Iva02, Theorem 7.1.E] and behaves similarly; both groups are torsion-free and passing to them avoids a good deal of troublesome periodic behaviour. In this vein, we will require the following consequence of [HM20, Theorem 3.1].

Proposition 2.5

[HW20], Lemma 2.6

Suppose that $G < \mathrm{{\mathrm{IA}}_{N}(\mathbb{Z}}/\mathrm{3\mathbb{Z})}$. If the G-orbit of $T\in {\mathcal{FS}}$ is finite, then G fixes T; moreover G<Stab⁰(T) and G fixes every collapse of T.

If T is a free splitting with one F_N-orbit of edges, we say that T is a one-edge splitting. A one-edge splitting is nonseparating if the quotient graph T/F_N is a loop, and separating otherwise. The link between free splittings and our study of direct products of free groups is the following extract from [HW20, Theorem 6.1]. Its original proof was in the context of direct products of free groups; we sketch below how the proof extends to commuting families of full-sized groups.

Theorem 2.6

[HW20], Theorem 6.1

Suppose $G < \mathrm{{\mathrm{IA}}_{N}(\mathbb{Z}}/\mathrm{3\mathbb{Z})}$ is generated by a set G₁,…,G_2N−4 of pairwise-commuting full-sized subgroups. Then G fixes a one-edge nonseparating free splitting of F_N.

Sketch proof

Let $\mathcal{F}$ be a maximal, proper, G-periodic free factor system. Then by [HM20, Theorem 3.1], as G is contained in $\mathrm{{\mathrm{IA}}_{N}(\mathbb{Z}}/\mathrm{3\mathbb{Z})}$, the free factor system $\mathcal{F}$ is fixed by G. A free factor system is sporadic if either $\mathcal{F}=\{[A]\}$ is given by a single factor of rank N−1 or $\mathcal{F}=\{[A],[B]\}$, where Rank(A)+Rank(B)=N. A system is nonspordadic if it is not sporadic.

If $\mathcal{F}$ is nonsporadic, then the relative free factor complex ${\mathrm{{FF}}}(F_{N},\mathcal{F})$ is hyperbolic and has infinite diameter, and the considerations in Sect. 4 of [HW20] apply equally well to commuting families as they do to direct products: either G has bounded orbits in ${\mathrm{{FF}}}(F_{N},\mathcal{F})$, or after reindexing the G_i, the subgroup generated by G₁,…,G_2N−5 has a finite orbit in the Gromov boundary of ${\mathrm{{FF}}}(F_{N},\mathcal{F})$ (this happens if the last subgroup G_2N−4 contains a loxodromic element Φ: as the rest of the G_i commute with Φ they preserve the endpoints of its axis). The group G cannot have bounded orbits as by [GH19, Proposition 5.1], G would have a finite orbit in ${\mathrm{{FF}}}(F_{N},\mathcal{F})$, contradicting maximality of $\mathcal{F}$. Furthermore, the subgroup generated by G₁,…,G_2N−5 cannot have a finite orbit in the boundary of ${\mathrm{{FF}}}(F_{N},\mathcal{F})$, as by [HW20, Lemma 6.5], the product rank of the stabilizer of any point on the Gromov boundary of ${\mathrm{{FF}}}(F_{N},\mathcal{F})$ is less than 2N−5.

We may therefore assume $\mathcal{F}$ is sporadic. In this case, $\mathcal{F}$ determines a G-invariant one-edge splitting T (there is a unique one-edge splitting where $\mathcal{F}$ is elliptic). It is shown in the proof of [HW20, Theorem 6.1] that the stabilizer of a separating one-edge splitting has product rank at most 2N−5, therefore T is nonseparating. □

2.4 Automorphic lifts

Subgroups that stabilize free splittings in Out(F_N) have the striking feature that they virtually lift to Aut(F_N) (see, for instance [HM23]). To describe this lifting, we need some further notation. Given G<Out(F_N), we let $\tilde{G}$ denote the preimage of G in Aut(F_N). We view F_N as a subgroup of $\tilde{G}$ via the identification g↦ad_g of F_N with the inner automorphisms. The following lemma is well known (see, for example, Lemma 6.7 of [BGH22]).

Lemma 2.7

Let T be a splitting of F_N such that the action of F_N on T is irreducible. A subgroup G<Out(F_N) fixes T if and only if the action of F_N on T extends to an action of $\tilde{G}$ on T.

Proof

As in Sect. 2.3, G fixes T if and only if for each $\phi \in \tilde{G}$ there exists a unique isomorphism f_ϕ:T→T such that f_ϕ(gx)=ϕ(g)f_ϕ(x). If the action $f:F_{N} \to \mathrm{{Isom}(T)}$ extends to an action $\tilde{f}: \tilde{G} \to \mathrm{{Isom}(T)}$ then $f_{\phi}:=\tilde{f}(\phi )$ suffices. Conversely, if G fixes T and $\phi , \psi \in \tilde{G}$ then the isomorphisms f_ϕ and f_ψ satisfy

$$\begin{aligned} f_{\phi }f_{\psi}(gx)&=f_{\phi}(\psi (g)f_{\psi}(x)) \\ &=\phi \psi (g)f_{\phi }f_{\psi}(x), \end{aligned}$$

so by uniqueness f_ϕf_ψ=f_ϕψ, and ϕ↦f_ϕ gives the required action. □

Understanding this extended action allows one to construct automorphic lifts.

Proposition 2.8

Existence of automorphic lifts

Let G be a subgroup of ${\mathrm{IA}}_{\mathrm{N}}(\mathbb{Z}/ \mathrm{3\mathbb{Z}})$ that fixes a free splitting T and let $\tilde{G}<{\mathrm{{Aut}}}(F_{N})$ be its preimage, which acts on T. For each edge e in T, the stabilizer $\tilde{G}_{e} < \tilde{G}$ of e is isomorphic to G. The isomorphism is given by the restriction to $\tilde{G}_{e}$ of the quotient map π:Aut(F_N)→Out(F_N).

Proof

As T is a free splitting, no inner automorphism fixes e and the map $\tilde{G}_{e} \to G$ is injective. To see that it is surjective, suppose Φ∈G and let ϕ∈Φ be a representative of Φ. Proposition 2.5 tells us that G<Stab⁰(T) and hence $\tilde{G}$ preserves the F_N-orbits of edges in T. Therefore f_ϕ(e)=ge for some g∈F_N and

$$ f_{{\mathrm{{ad}}}_{g}^{-1}\phi}(e)=f_{{\mathrm{{ad}}}_{g}^{-1}}f_{\phi}(e)=g^{-1} \cdot ge=e. $$

It follows that ${\mathrm{{ad}}}_{g}^{-1}\phi$ is a representative of Φ in $\tilde{G}_{ e}$. □

3 Direct products of free groups in M _k(F ₂) with maximal product rank

We consider semidirect products of the form M_k(A)=(A×⋯×A)⋊Aut(A), where there are k copies of the arbitrary group A and the action of Aut(A) is diagonal. Writing elements of M_k(A) in the form (g₁,…,g_k;ϕ), the group operation is

$$ (g_{1},\dots ,g_{k} ; \phi ).(h_{1},\dots ,h_{k} ; \psi ) = (g_{1} \phi (h_{1}),\dots ,g_{k}\phi (h_{k}) ; \phi \circ \psi ). $$

These groups arise naturally in many settings. For example, if X=K(A,1) then the group of homotopy classes of homotopy equivalences X→X fixing (k+1) marked points is isomorphic to M_k(A). When A is free, these groups play an important role in the study of graph cohomology [C+16] and homology stability results for automorphism groups of free groups [HV04]. We shall be concerned almost entirely with the case A=F₂.

Our interest in M_k(F₂) stems from the fact that if T is the Bass–Serre tree of a collapsed rose with N−2 petals in the boundary of Outer space then, as we will see later, Stab⁰(T)≅M_2N−5(F₂). Our purpose in this section is to give a detailed description of the ways in which a direct product of (k+1) nonabelian free groups can be embedded in M_k(F₂).

3.1 Generalities

We distinguish between the k visible copies of A in M_k(A) by writing M_k(A)=(A₁×⋯×A_k)⋊Aut(A). If A has trivial centre, then g↦ad_g defines an isomorphism from A to the group of inner automorphisms Inn(A)<Aut(A). We claim that this gives rise to a natural embedding of the direct product A^k+1↪M_k(A) with image

$$ M^{0}_{k}(A) := (A_{1}\times \cdots \times A_{k})\rtimes {\mathrm{{Inn}}}(A), $$

where the last summand in A^k+1 maps to

$$ J := \{(g^{-1},\dots ,g^{-1}; {\mathrm{{ad}}}_{g}) \mid g\in A\} < M_{k}(A). $$

The existence of this embedding points to the fact A₁×⋯×A_k<M_k(A) is not a characteristic subgroup, and the semidirect product decomposition defining M_k(A) is less canonical than the short exact sequence

$$ 1\to M^{0}_{k}(A) \to M_{k}(A) \to {{\mathrm{{Out}}}}(A) \to 1. $$

Lemma 3.1

With the notation established above,

• 1. (g₁,…,g_k,x)↦(g₁x⁻¹,…,g_kx⁻¹;ad_x) defines an isomorphism $A^{k+1}\overset{\cong}{\to}M^{0}_{k}(A)$ with inverse (g₁,…,g_k;ad_x)↦(g₁x,…,g_kx,x).

• 2. For i=1,…,k, there exists an involution α_i∈Aut(M_k(A)) that exchanges A_i and J while restricting to the identity on Aut(A) and A_ℓ≠i;

• 3. these α_i generate a copy of the symmetric group Sym(k+1)↪Aut(M_k(A)), and the embedding $A^{k+1}\to M^{0}_{k}(A)< M_{k}(A)$ is Sym(k+1)-equivariant.

Proof

For (1): it is clear that these maps are mutually inverse and a straightforward calculation establishes that they are homomorphisms.

To prove (2) and (3), one verifies that the formula

$$ \alpha _{i} : (g_{1},\dots ,g_{k}; \phi ) \mapsto (g_{1}g_{i}^{-1}, \dots ,g_{i-1}g_{i}^{-1},\, g_{i}^{-1},\, g_{i+1}g_{i}^{-1},\dots ,g_{k}g_{i}^{-1}; {\mathrm{{ad}}}_{g_{i}}\phi ) $$

defines an automorphism with the desired properties. □

3.2 Fixed subgroups, centralisers, and Nielsen transformations

We remind the reader that an automorphism τ∈Aut(F₂) is called a Nielsen transformation if there is a basis {x₁,x₂} of F₂ such that

$$ \tau (x_{1})=x_{1}x_{2} \text{ and } \tau (x_{2})=x_{2}. $$

(2)

Any two Nielsen transformations are conjugate in Aut(F₂) and there are various ways to distinguish them from other automorphisms. In ${\mathrm{{Out}}}(F_{2})\cong {\mathrm{{GL}}}(2,\mathbb{Z})$, an automorphism represents a power of a Nielsen transformation if and only if its associated matrix has trace 2 and determinant 1.

Lemma 3.2

If a subgroup H<Aut(F₂) consists entirely of powers of Nielsen transformations, then it is cyclic.

Proof

H intersects Inn(F₂) trivially and therefore injects into ${\mathrm{{Out}}}(F_{2})\cong {\mathrm{{GL}}}(2,\mathbb{Z})$, where its image consists entirely of elements that have trace 2 and determinant 1; let M be such an element. We choose a basis so that the image of H contains $E_{n}= \begin{pmatrix} 1 & n \cr 0 & 1 \end{pmatrix}$ for some non-zero integer n. If $M= \begin{pmatrix} a & b \cr c & d \end{pmatrix}$, then ME_n has trace 2 only if c=0 and a=d=1. Hence the image of H is contained in the cyclic subgroup of ${\mathrm{{GL}}}(2,\mathbb{Z})$ generated by E₁. □

A characterisation of Nielsen transformations that will be useful to us here concerns the rank of fixed subgroups; this is due to Collins and Turner [CT96]. Bestvina and Handel’s solution to the Scott Conjecture [BH92] shows that the subgroup of Fix(ϕ)<F_N fixed by an automorphism ϕ has rank at most N. Theorem A of [CT96] gives a complete description of the automorphisms with rk(Fix(ϕ))=N; in the case N=2, these automorphisms are the powers of Nielsen transformations.

Proposition 3.3

[CT96], Theorem A

If ϕ∈Aut(F₂) fixes a non-cyclic subgroup of F₂, then ϕ is a power of a Nielsen transformation. If τ is the Nielsen automorphism given in Equation (2) then the fixed subgroup of every nontrivial power of τ is $\langle x_{1}x_{2}x_{1}^{-1}, x_{2} \rangle$.

Corollary 3.4

• 1. If Λ<Inn(F₂) is not cyclic, then its centraliser C(Λ)<Aut(F₂) is either trivial or else the cyclic subgroup generated by a Nielsen transformation.

• 2. Let τ₀,τ₁∈Aut(F₂) be Nielsen transformations. If 〈τ₀〉∩〈τ₁〉≠1, then $\tau _{0}=\tau _{1}^{\pm 1}$.

• 3. Let ϕ,ψ∈Aut(F₂), not both trivial. If Fix(ϕ)∩Fix(ψ) is not cyclic, then there is a unique Nielsen transformation τ^±1 such that ϕ,ψ∈〈τ〉.

• 4. rk_F(Aut(F₂))=1.

Proof

This proof is a variation on the argument used by Gordon [Gor04, Theorem 3.2] to prove that rk_F(Aut(F₂))=1. If ϕ∈Aut(F₂) centralizes Λ, then ϕ(w)=w for all ad_w∈Λ, so ϕ is a power of a Nielsen transformation. Thus C(Λ) consists entirely of powers of Nielsen transformations, so it is cyclic, by Lemma 3.2. If τ is a Nielsen transformation then Fix(τ)=Fix(τ^p) for p≠0, so τ^p∈C(Λ) implies τ∈C(Λ). This proves (1).

For (2), using again the fact that Fix(τ)=Fix(τ^p) for all p≠0 we see that Fix(τ₀)=Fix(τ₁). Then apply (1) with Λ={ad_w:w∈Fix(τ₀)}, noting that a Nielsen transformation generates any cyclic subgroup to which it belongs. The existence of τ in (3) is proved by applying (1) to {ad_x∣x∈Fix(ϕ)∩Fix(ψ)}. Uniqueness follows immediately from (2).

For (4), since Out(F₂) is virtually free, the centraliser of any nonabelian free subgroup is finite. So if there were a copy of F₂×F₂ in Aut(F₂) then at least one of the factors, say F₂×1, would have to intersect Inn(F₂) non-trivially. The intersection would be normal in F₂×1, hence non-cyclic. On the other hand, the centraliser of this intersection would contain 1×F₂, contradicting (1). □

Lemma 3.5

Let τ₀,τ₁∈Aut(F₂) be Nielsen transformations. If some non-zero powers of τ₀ and τ₁ commute, then τ₀ and τ₁ commute, and $\tau _{1}= {\mathrm{{ad}}}_{w}\tau _{0}^{\pm 1}$ for some w∈Fix(τ₀).

Proof

By hypothesis, $[\tau _{0}^{p},\tau _{1}^{q}]=1$ for some p,q≠0, that is $(\tau _{0}^{-p} \tau _{1}\tau _{0}^{p})^{q}=\tau _{1}^{q}$. From Corollary 3.4(2) we deduce $\tau _{0}^{-p} \tau _{1}\tau _{0}^{p}=\tau _{1}$. Similarly, $(\tau _{1}^{-1} \tau _{0}\tau _{1})^{p}=\tau _{0}^{p}$ implies $\tau _{1}^{-1}\tau _{0}\tau _{1}=\tau _{0}$.

In the linear action of ${\mathrm{{GL}}}(2,\mathbb{Z})\cong {\mathrm{{Out}}}(F_{2})$ on $\mathbb{R}^{2}$, the image of each Nielsen transformation τ has a unique eigenspace V and τ generates the pointwise stabiliser of V in ${\mathrm{{SL}}}(2,\mathbb{Z})$. As τ₀ and τ₁ commute, their eigenspaces coincide. Thus, replacing τ₀ by its inverse if necessary, τ₁=ad_wτ₀. Comparing $\tau _{0}\tau _{1} = {\mathrm{{ad}}}_{\tau _{0}(w)} \tau _{0}^{2}$ to $\tau _{1}\tau _{0} = {\mathrm{{ad}}}_{{w}} \tau _{0}^{2}$, we have τ₀(w)=w. □

We remind the reader that two elements ϕ,ψ∈Aut(F_N) are defined to be similar if there is an inner automorphism ad_y such that $\phi = {\mathrm{{ad}}}_{y}\circ \psi \circ {\mathrm{{ad}}}_{y}^{-1} = {\mathrm{{ad}}}_{y \psi (y)^{-1}} \circ \psi$. As conjugacy preserves the set of (powers of) Nielsen transformations, so does similarity.

Lemma 3.6

Let τ∈Aut(F₂) be a Nielsen transformation and suppose p≠0 and w∈F₂. Then ϕ=ad_wτ^p is a power of a Nielsen transformation if and only if w=yτ^p(y)⁻¹ for some y∈F₂. Equivalently, ϕ=ad_wτ^p is a power of a Nielsen transformation if and only if there exists y∈F₂ such that $\phi = ({\mathrm{{ad}}}_{y}\circ \tau \circ {\mathrm{{ad}}}_{y}^{-1})^{p}$.

Proof

In the light of Proposition 3.3, it suffices to argue that Fix(ϕ) is noncyclic if and only if ϕ is similar to τ^p. This follows from the strengthened form of the Scott Conjecture proved in [BH92, Corollary 6.4], which says that if ϕ∈Aut(F_N) and if S is a set of representatives of the similarity classes of ϕ in its outer class [ϕ], then

$$ \sum _{\psi \in S}\max \{({\mathrm{{Rank}}}({\mathrm{{Fix}}}(\psi ))-1),0\} \leq N-1. $$

In particular, there are only finitely many similarity classes in any outer class that have noncyclic fixed subgroups, and when N=2 there can be at most one. Hence any automorphism in [τ] with a noncyclic fixed subgroup is similar to τ. □

3.3 A description of subgroups in M _k(F ₂) with maximal product rank

In the remainder of this section, we focus on the case A=F₂. To lighten the notation, we make the abbreviations M_k=M_k(F₂) and F^k:=(A₁×⋯×A_k). We have a fixed identification of each A_i with F₂, with respect to which the action defining the semidirect product M_k=F^k⋊Aut(F₂) is diagonal. Note that, for each index i, the subgroup $A_{i}^{\tau }< A_{i}$ fixed by τ∈Aut(A_i)=Aut(F₂) is the intersection of A_i<F^k with the centraliser in M_k of ${\underline{\tau }}=(1,\dots ,1;\tau )$. Likewise, we define J^τ<J to be the tuples of the form (g⁻¹,…,g⁻¹;ad_g) with g∈Fix(τ). Then J^τ is defined to be the intersection of J (as defined in Sect. 3.1) with the centraliser of ${\underline{\tau }}$.

Remark 3.7

As rk_F(Aut(F₂))=1 and product rank is subadditive with respect to exact sequences (Lemma 2.2), the exact sequence $1 \to F_{2}^{k} \to M_{k}(F_{2}) \to {\mathrm{{Aut}}}(F_{2}) \to 1$ implies that rk_F(M_k(F₂))≤k+1. The discussion in Sect. 3.1 gives an embedding of $F_{2}^{k+1}$ in M_k(F₂), showing that the product rank is exactly k+1.

We want to classify embeddings D↪M_k, where

$$ D= \langle D_{1},\ldots , D_{k+1} \rangle $$

is generated by a collection of pairwise commuting subgroups D_i such that each D_i is full-sized. (The D_i need not be finitely generated.) We will be led to consider the images of the subgroups D_i under the retraction π:M_k→Aut(F₂) and the associated projection ${\overline{\pi }}: M_{k}\to {\mathrm{{Out}}}(F_{2})$. Let $K_{i}:=\ker \pi |_{D_{i}}$.

Theorem 3.8

With the notation established above:

• 1. After permuting the indices, [D_i,D_i]<A_i for i=1,…,k and [D_k+1,D_k+1]<J;

• 2. if the centraliser $C_{M_{k}}(D) < M_{k}$ is non-trivial then there is a Nielsen transformation τ such that $C_{M_{k}}(D) =\langle {\underline{\tau }}\rangle$, and after conjugating by an element of F^k×J,

  $$ D< A_{1}^{\tau}\times \cdots \times A_{k}^{\tau}\times J^{\tau } \times \langle {\underline{\tau }}\rangle , $$

  with $D_{i}< A_{i}^{\tau}\times \langle {\underline{\tau }}\rangle$ for i=1,…,k and $D_{k+1}< J^{\tau}\times \langle {\underline{\tau }}\rangle$;

• 3. if $C_{M_{k}}(D)$ is trivial, then D satisfies one of the following conclusions, after conjugating by an element of F^k×J:

  i.:

  ${\overline{\pi }}(D)=1$ and D_i<A_i for i=1,…,k, while D_k+1<J – in particular,

  $$ D< A_{1}\times \cdots \times A_{k}\times J; $$

  ii.:

  ${\overline{\pi }}(D_{i})=1$ for i=1,…,k but ${\overline{\pi }}(D_{k+1})\neq 1$: in this case there is a Nielsen transformation τ such that $D_{i}< A_{i}^{\tau}$ for i=1,…,k and $D_{k+1}< \langle J, {\underline{\tau }}\rangle$, so

  $$ D< A_{1}^{\tau}\times \cdots \times A_{k}^{\tau}\times \langle J,{ \underline{\tau }}\rangle ; $$

  iii.:

  ${\overline{\pi }}(D_{k+1})= 1$ but there is a unique j≤k such that ${\overline{\pi }}(D_{j})\neq 1$: in this case $D_{j}<\langle A_{j}, {\underline{\tau }}\rangle$ while D_k+1<J^τ and $D_{i}< A_{i}^{\tau}$ for j≠i≤k, whence

  $$ D< A_{1}^{\tau}\times \cdots \times A_{j-1}^{\tau}\times \langle A_{j},{ \underline{\tau }}\rangle \times A_{j+1}^{\tau}\dots \times A_{k}^{ \tau}\times J^{\tau}. $$

The following implicit feature of the theorem warrants explicit mention.

Addendum 3.9

If ${\overline{\pi }}(D_{i})$ is non-trivial for at least two of the subgroups D_i<D, then $C_{M_{k}}(D)=\langle {\underline{\tau }}\rangle$ for some Nielsen transformation τ.

Remarks 3.10

• 1. The action of α_j∈Aut(M_k) interchanges cases 3(ii) and 3(iii).

• 2. If $C_{M_{k}}(D)$ is non-trivial, D might still conform to one of the descriptions in 3(i-iii).

• 3. In cases 3(ii) and 3(iii), the subgroups $\langle A_{j},{\underline{\tau }}\rangle$ and $\langle J,{\underline{\tau }}\rangle$ are isomorphic to $F_{2}\rtimes _{\tau}\mathbb{Z}$, a virtually special 3-manifold group in which free groups abound.

3.4 The proof of Theorem 3.8

Lemma 3.11

Consider D=〈D₁,…,D_k+1〉↪M_k=F^k⋊Aut(F₂) with each D_i full-sized and [D_i,D_j]=1 for all 1≤i<j≤k+1.

• 1. There is a unique index i₀ such that $\pi |_{D_{i_{0}}}$ is injective.

• 2. There is a Nielsen transformation τ such that π(D_j)<〈τ〉 for all j≠i₀, and

• 3. ${\overline{\pi }}(D_{i_{0}})< \langle [\tau ]\rangle$.

• 4. $\pi |_{D_{i_{0}}}^{-1}({\mathrm{{Inn}}}(F_{2})) < J$.

Proof

Let $D_{j}'< D_{j}$ be a free nonabelian subgroup. As in the proof of Proposition 2.3, the direct product $D_{1}'\times D_{2}'\times \cdots \times D_{k}'$ embeds into D. Let $K_{j}=\ker \pi |_{D_{j}}$ and let $K_{j}'=\ker \pi |_{D_{j}'}$. If $K_{j}'$ is nontrivial then it is a nonabelian free group. Since $K_{1}'\times \cdots \times K_{k+1}'< F^{k}$ and rk_F(F^k)=k, it follows that $\pi |_{D_{j}'}$ is injective for at least one index i₀. As rk_F(Aut(F₂))=1, this index i₀ is unique. We claim that $\pi |_{D_{i_{0}}}$ is also injective. Indeed, the remaining K_j generate a subgroup of F^k isomorphic to a direct product of k nonabelian free groups, which has a trivial centralizer in F^k. As $K_{i_{0}}$ is contained in this centralizer it is also trivial.

To lighten the notation, we assume i₀=k+1. Then K=K₁×K₂×⋯×K_k<F^k is a direct product of k nonabelian free groups, and we can relabel the indices to assume K_j<A_j for j=1,…,k. We fix a pair of non-commuting elements ${\underline{u}}_{j},{\underline{v}}_{j}\in K_{j}$ for j=1,…,k and denote their non-trivial coordinates by u_j and v_j respectively; for example, ${\underline{u}}_{1}=(u_{1},1\dots ,1;1)$ and ${\underline{v}}_{1}=(v_{1},1\dots ,1;1)$.

The remaining parts of the lemma require an analysis of the elements of D_k+1 – let g=(ω₁,…,ω_k;ϕ) be such. Since D_k+1 commutes with K₁, we have $g{\underline{u}}_{1} = {\underline{u}}_{1} g$, hence

$$ (\omega _{1} \phi (u_{1}), \omega _{2},\dots , \omega _{k}; \phi ) = (u_{1} \omega _{1} , \omega _{2},\dots , \omega _{k}; \phi ), $$

whence $\omega _{1} \phi (u_{1}) \omega _{1}^{-1} = u_{1}$. Similarly, $\omega _{1} \phi (v_{1}) \omega _{1}^{-1} = v_{1}$. Since u₁ and v₁ do not commute, ${\mathrm{{Fix}}}({\mathrm{{ad}}}_{\omega _{1}}\circ \phi )$ is not cyclic. Therefore, by Proposition 3.3, either $\phi ={\mathrm{{ad}}}_{\omega _{1}}^{-1}$ or else ${\mathrm{{ad}}}_{\omega _{1}}\circ \phi$ is a non-zero power of a Nielsen transformation. In the latter case, Lemma 3.6 tells us that ϕ itself must be a power of a Nielsen transformation, say $\phi =\tau _{0}^{p}$, and $\omega _{1}=y_{1}\tau _{0}^{p}(y_{1})^{-1}$ for some y₁∈F₂.

Repeating this argument with {u_j,v_j} in place of {u₁,v₁}, we see that either $\phi ={\mathrm{{ad}}}_{\omega _{j}}^{-1}$ for j=1,…,k or else $\phi =\tau _{0}^{p}$ and ω_j has the form $y_{j}\tau _{0}^{p}(y_{j})^{-1}$ for j=1,…,k. From the former case we deduce that

$$ \pi |_{D_{k+1}}^{-1}({\mathrm{{Inn}}}(F_{2})) < J, $$

(3)

as claimed in (4). From the latter case we deduce that ${\overline{\pi }}(D_{k+1})<{\mathrm{{Out}}}(F_{2})$ consists entirely of elements of trace 2 and determinant one. As in Lemma 3.2, this implies that ${\overline{\pi }}(D_{k+1})<{\mathrm{{Out}}}(F_{2})$ is cyclic, generated by $[\tau _{0}^{r}]$, say, where $\tau _{0}^{r}\in \pi (D_{k+1})$.

At this stage, we know that π(D_k+1)<Aut(F₂) is full-sized and ${\overline{\pi }}(D_{k+1})<{\mathrm{{Out}}}(F_{2})$ is cyclic. Thus Λ:=π(D_k+1)∩Inn(F₂) is not abelian, and Corollary 3.4 provides a Nielsen transformation τ that generates the centralizer of Λ in Aut(F₂). As π(D_j) commutes with Λ when j≤k, part (2) of the lemma is proved. (If π(D_j)=1 for all j≤k, then we take τ=τ₀.)

If π(D_j)=〈τ^p〉 for some j≤k and p≠0, then τ^p∈π(D_j) commutes with $\tau _{0}^{r}\in \pi (D_{k+1})$. Hence τ₀=ad_w∘τ^±1, by Lemma 3.5, and (3) is proved. □

Proof of Theorem 3.8

With the lemma in hand, we may assume that for j≤k we have D_j=K_j<A_j or D_j=K_j⋊〈T_j〉 with K_j<A_j nonabelian and $T_{j}=(t_{j1},\dots , t_{jk} ; \tau ^{p_{j}})$, some p_j≠0. Also, D_k+1<J or else D_k+1=K₀⋊〈T₀〉 with K₀<J and $T_{0}=(t_{01},\dots , t_{0k} ; \tau _{0}^{p_{0}})$. Part (1) of the theorem follows. Observe that C(D) is contained in the centralizer of K₁×K₂×⋯×K_k, so intersects A₁×A₂×⋯×A_k trivially. Hence C(D)<1⋊Aut(F₂). This subgroup of Aut(F₂) will fix the nonabelian subgroup K₁⊂A₁≅F₂ in the natural action, so is trivial or else contained in the subgroup generated by a Nielsen automorphism, as in Corollary 3.4(1).

Let us first assume that there are at least two indices with D_j=K_j⋊〈T_j〉 and prove that this forces D to be as described in part (2) of the theorem. For clarity of exposition, we assume that this set of indices includes {1,2}. (The superficially-exceptional case where one of the indices is (k+1) is reduced to this case by applying one of the automorphisms α_i.)

If j≠1, then for every ${\underline{w}}\in K_{j}$ we have $T_{1}{\underline{w}}= {\underline{w}}T_{1}$. The j-coordinate of $T_{1} {\underline{w}}$ is $t_{1j}\tau ^{p_{1}}(w)$, whereas the j-coordinate of ${\underline{w}}T_{1}$ is wt_1j. Thus $\tau ^{p_{1}}(w) = t_{1j}^{-1}w t_{1j}$ for all w∈K_j; in other words $f_{1j}:={\mathrm{{ad}}}_{t_{1j}}\tau ^{p_{1}}$ fixes K_j.

Lemma 3.6 provides y_1j∈F₂ such that

$$ t_{1j} = y_{1j}\,\tau ^{p_{1}}(y_{1j})^{-1}, $$

so $f_{1j}={\mathrm{{ad}}}_{y_{1j}} \circ \tau ^{p_{1}} \circ {\mathrm{{ad}}}_{y_{1j}}^{-1}$. Similarly, for j≠2 we obtain $f_{2j}={\mathrm{{ad}}}_{y_{2j}} \circ \tau ^{p_{2}} \circ {\mathrm{{ad}}}_{y_{2j}}^{-1}$ fixing K_j. Corollary 3.4(3) tells us that f_1j and f_2j are powers of a common Nielsen transformation, hence

$$ {\mathrm{{ad}}}_{y_{1j}} \circ \tau \circ {\mathrm{{ad}}}_{y_{1j}}^{-1} = {\mathrm{{ad}}}_{y_{2j}} \circ \tau \circ {\mathrm{{ad}}}_{y_{2j}}^{-1}, $$

with $y_{1j}y_{2j}^{-1} \in {\mathrm{{Fix}}}(\tau )$ (note that by examining the images of these automorphisms in Out(F_N), we cannot have τ on one side and τ⁻¹ on the other side of this equation). Taking powers and simplifying, this implies that for all $m\in \mathbb{Z}$,

$$ y_{1j}\tau ^{m}(y_{1j})^{-1} = y_{2j}\tau ^{m}(y_{2j})^{-1}. $$

(4)

We now conjugate D by γ=(y₂₁,y₁₂,y₁₃,…,y_1k;1)⁻¹, noting that

$$ (y_{21}, y_{12}, y_{13},\dots , y_{1k}; 1)^{-1} \ T_{1} \ (y_{21}, y_{12}, y_{13},\dots , y_{1k}; 1) = (*, z_{12}, z_{13},\dots ,z_{1k}; \tau ^{p_{1}}), $$

where for j≥2 we have

$$ z_{1j} = y_{1j}^{-1} t_{1j} \tau ^{p_{1}}(y_{1j}) = y_{1j}^{-1} y_{1j} \tau ^{p_{1}}(y_{1j})^{-1} \tau ^{p_{1}}(y_{1j}) =1. $$

Likewise,

$$ (y_{21}, y_{12}, y_{13},\dots , y_{1k}; 1)^{-1} \ T_{2} \ (y_{21}, y_{12}, y_{13},\dots , y_{1k}; 1) = (z_{21}, *, z_{23},\dots ,z_{2k}; \tau ^{p_{2}}), $$

where, using equation (4) to replace $y_{2j} \tau ^{p_{2}}(y_{2j})^{-1}$ by $y_{1j} \tau ^{p_{2}}(y_{1j})^{-1}$, for j≠1,2 we have

$$ z_{2j} = y_{1j}^{-1} t_{2j} \tau ^{p_{2}}(y_{1j}) = y_{1j}^{-1} y_{2j} \tau ^{p_{2}}(y_{2j})^{-1} \tau ^{p_{2}}(y_{1j}) = y_{1j}^{-1} y_{1j} \tau ^{p_{2}}(y_{1j})^{-1} \tau ^{p_{2}}(y_{1j}) =1. $$

An entirely similar argument applies to each index j with D_j=K_j⋊〈T_j〉.

Thus, after this conjugation (and abusing notation by identifying D with D^γ), the T_j have the form

$$ T_{1} = (t_{1},1,\dots ,1;\tau ^{p_{1}}),\ \ T_{2} = (1,t_{2},1,\dots ,1;\tau ^{p_{2}}),\ \ T_{j} = (1,\dots ,t_{j},\dots ,1; \tau ^{p_{j}}). $$

The commutation [T₁,K_j]=1 now forces $K_{j}<{\mathrm{{Fix}}}(\tau ^{p_{1}})={\mathrm{{Fix}}}( \tau )$ for j>1 (including the case K_j=D_j). And [T₂,K₁]=1 forces K₁<Fix(τ). Finally, the relations [T₁,T_j]=1 imply t_j∈Fix(τ) for j>1. In particular, $T_{j} \in A_{j}^{\tau }\times \langle {\underline{\tau }}\rangle$, hence $D_{j} < A_{j}^{\tau }\times \langle {\underline{\tau }}\rangle$ for j≤k. (The superficially-exceptional case j=k+1 can again be handled by exchanging D_k+1 and some D_j using α_j∈Aut(M_k).) This shows that when at least two of the T_j are nontrivial, the D_j are as described in (2). In particular, ${\underline{\tau }}$ centralizes D. As C(D) is contained in a cyclic subgroup as described at the start of this proof, we conclude that $C(D)=\langle {\underline{\tau }}\rangle$.

It remains to consider what happens when ${\overline{\pi }}(D_{j})\neq 1$ for at most one index j. Case 3(i) is covered by Lemma 3.11 and Case 3(ii) can be reduced to 3(iii) by applying the automorphism α_j, so we address Case 3(iii), taking j=1 for clarity. The proof in this case is a simplified version of the proof of (2): we have D_j<A_j for j=2,…,k and D_k+1<J, and after conjugating we may assume that D₁<A₁⋊〈T₁〉 where T₁=(t₁,1,…,1;τ^p) with p≠0. Again, the commutation [T₁,D_j]=1 forces $D_{j}< A_{j}^{\tau}$ for 2≤j≤k and D_k+1<J^τ.

In order to complete the proof, we have to argue that if C(D) is nontrivial, then D is as described in (2). Suppose C(D) contains a nontrivial element ζ. Let $D_{j}^{+}=\langle D_{j}, \zeta \rangle$, and let D⁺ be the subgroup of M_k generated by the pairwise-commuting family $D_{1}^{+}, D_{2}^{+}, \ldots , D_{k+1}^{+}$. The centre of each $D_{j}^{+}$ is nontrivial, so $D_{j}^{+}$ is of the form $D_{j}^{+}=K_{j}^{+} \rtimes \langle T_{j}^{+} \rangle$ and hence projects nontrivially to Aut(F₂). We have proved that in this case, D⁺ is as described in (2), so $D_{j} < D_{j}^{+} < A_{j}^{\tau }\times \langle {\underline{\tau }} \rangle$ and $D_{k+1} < D_{k+1}^{+} < J^{\tau }\times \langle {\underline{\tau }} \rangle$. Finally, since ${\underline{\tau }}$ visibly centralizes D, and the centralizer of D is contained in a cyclic subgroup generated by a Nielsen transformation, $C(D)=\langle {\underline{\tau }}\rangle$. □

3.5 Extending by Sym(k+1)

Theorem 3.8 describes embeddings into M_k of groups generated by a commuting family of (k+1) full-sized groups. The following proposition shows that one gets no extra embeddings when the target is enlarged to M_k⋊Sym(k+1), where the action in the semidirect product is the same as Lemma 3.1: the transposition (i k+1)∈Sym(k+1) acts as α_i.

Proposition 3.12

If D is generated by a set of (k+1) pairwise-commuting full-sized subgroups, then the image of every embedding D↪M_k⋊Sym(k+1) lies in M_k×1. Furthermore, if ϕ∈M_k⋊Sym(k+1) centralizes D, then ϕ lies in M_k×1.

Proof

We identify D with its image in M_k⋊Sym(k+1). There is no loss of generality in assuming that D is finitely generated. Let D^∗<D be the subgroup obtained by replacing each subgroup D_i<D with the intersection of the kernels of all non-trivial homomorphisms from D_i to (the abstract group) Sym(k+1). Then D^∗<M_k×1 and as in Theorem 3.8(1) we may assume that $[D^{*}_{i}, D^{*}_{i}]< A_{i}$ for i≤k and $[D^{*}_{k+1}, D^{*}_{k+1}]< J$. The action of D by conjugation on D^∗ preserves each of the subgroups $[D^{*}_{i}, D^{*}_{i}]$. In contrast, conjugation in M_k⋊Sym(k+1) by any element of the form (m;σ) will (if we write J=A_k+1) send A_i to A_σ(i); in particular it will not leave $[D^{*}_{i}, D^{*}_{i}]$ invariant if σ(i)≠i. If ϕ centralizes D then ϕ will also leave the groups $[D_{i}^{*},D_{i}^{*}]$ invariant, so must also lie in M_k×1. □

4 Stabilizers of collapsed roses and cages

In this paper we restrict our attention to two simple examples of free splittings, which are collapsed roses and cages. We will make use of Bass–Serre theory, for which the standard references are Serre’s book [Ser80] and the topological approach given by Scott and Wall [SW79].

4.1 The Bass–Serre tree of a collapsed rose

A collapsed rose with k petals is a graph of groups decomposition of F_N with a single vertex group and k loops with trivial edge groups. The vertex group is necessarily isomorphic to a free factor A≅F_N−k. We abuse notation slightly and refer to a free splitting T of F_N as a collapsed rose if the corresponding graph of groups is a collapsed rose. In the above notation, the vertex stabilizers of T are the conjugates of the free factor A. A rose with one loop is simply a nonseparating free splitting.

Let b_A be the vertex of T with stabilizer A and pick representatives e₁,…,e_k of each orbit of edges that have initial vertex b_A. A stable letter for e_i is a choice of element x_i such that x_ib_A is the terminal vertex of e_i. Changing either the representative e_i or the translating element gives possible stable letters of the form $ux_{i}^{\pm 1}v$, where u,v∈A. The free group F_N is generated by A and the stable letters x₁,…,x_k.

4.2 Stabilizers of collapsed roses in Out(F _N)

Let T be a splitting of F_N corresponding to a collapsed rose with k petals. As in Sect. 2.3, we let Stab(T) denote the stabilizer of T in Out(F_N). Letting x₁,…,x_k be a choice of stable letters for the rose, there is a finite subgroup W_k generated by automorphisms σ such that σ is the identity when restricted to A and for every stable letter $\sigma (x_{i})=x_{j}^{\epsilon}$ for some j and ϵ∈{1,−1}. One can think of W_k as the subgroup of Out(F_N) given by permuting and inverting the petals of the rose, and W_k is isomorphic to the semidirect product of $(\mathbb{Z}/2\mathbb{Z})^{k}$ and the symmetric group Sym(k). The homomorphism

$$ F:{\mathrm{Stab}}(T) \to {\mathrm{{Aut}}}(T/F_{N}) $$

to the automorphism group of the rose is split surjective, and W_k is a set of coset representatives of Stab⁰(T) in Stab(T).

Proposition 4.1

Let T be a collapsed rose with k petals. Let $\widetilde{\mathrm{{Stab}}}(T)$ and $\widetilde{\mathrm{{Stab}}}^{0}(T)$ be the respective preimages of Stab(T) and Stab⁰(T) in Aut(F_N). Let b_A be a vertex of the tree with F_N-stabilizer A and let $\widetilde{\mathrm{{Stab}}}(T)_{A}$ and $\widetilde{\mathrm{{Stab}}}^{0}(T)_{A}$ be the respective subgroups of $\widetilde{\mathrm{{Stab}}}(T)$ and $\widetilde{\mathrm{{Stab}}}^{0}(T)$ that fix b_A. An automorphism ϕ is an element of $\widetilde{\mathrm{{Stab}}}(T)_{A}$ if and only if there exists w∈W_k such that:

• 1. ϕ restricts to an automorphism of A and,

• 2. for each stable letter x_i there exist u_i,v_i∈A such that ϕ(x_i)=u_iw(x_i)v_i.

The group $\widetilde{\mathrm{{Stab}}}(T)_{A}$ is isomorphic to

$$ M_{2k}(A) \rtimes W_{k} =(A^{2k}\rtimes {\mathrm{{Aut}}}(A)) \rtimes W_{k}, $$

via the isomorphism

$$ \phi \mapsto (u_{1}^{-1}, \ldots , u_{k}^{-1}, v_{1}, \ldots , v_{k}; \phi |_{A} ; w). $$

The group W_k acts on M_2k(A) as a subgroup of the group Sym(2k+1) defined in Lemma 3.1, and $\phi \in \widetilde{\mathrm{{Stab}}}^{0}(T)_{A}$ if and only if w=1 in the above.

If e_j is the edge joining b_A to x_jb_A then the subgroup $\widetilde{\mathrm{{Stab}}}^{0}(T)_{e_{j}} < \widetilde{\mathrm{{Stab}}}^{0}(T)_{A}$ fixing e_j consists of automorphisms $\phi \in \widetilde{\mathrm{{Stab}}}^{0}(T)_{A}$ such that ϕ(x_j)=x_jv_j (i.e. u_j=1); it is isomorphic to M_2k−1(A)=A^2k−1⋊Aut(A).

Proof

By the work on automorphic lifts in Sect. 2.4, $\phi \in \widetilde{\mathrm{{Stab}}}(T)$ if and only if there exists a ϕ-twistedly equivariant map f_ϕ:T→T that preserves the F_N-orbits of edges and their orientations. Suppose that ϕ satisfies conditions (1) and (2) of the proposition. We define f_ϕ on the vertex set of T by the map of cosets gA↦ϕ(g)A (as the vertices of T correspond to cosets of A via Bass–Serre theory). Let $X=\{x_{1},\ldots , x_{k},x_{1}^{-1},\ldots , x_{k}^{-1}\}$. The cosets gA adjacent to 1A are of the form axA, for a∈A and x∈X. Therefore, under this correspondence between vertices and cosets of A, two vertices gA and hA span an edge in T if and only if g⁻¹h∈AXA. As ϕ(AXA)=AXA, the map f_ϕ preserves edges and so determines an isomorphism of T which is clearly ϕ-twistedly equivariant. Conversely, if $\phi \in \widetilde{\mathrm{{Stab}}}(T)_{A}$ then let f_ϕ:T→T be a ϕ-twistedly equivariant map fixing b_A. As f_ϕ(b_A)=b_A, it follows that ϕ preserves Stab(b_A)=A, so restricts to an automorphism of A. As f_ϕ preserves edges, using the same reasoning as above, we must have ϕ(AXA)=AXA, from which it follows that ϕ must also satisfy (2).

The action of ϕ on the quotient graph T/F_N is given by the last coordinate w∈W_k in its decomposition, so $\phi \in \widetilde{\mathrm{{Stab}}}^{0}(T)_{A}$ if and only if w=1. This gives an identification of $\widetilde{\mathrm{{Stab}}}^{0}(T)_{A}$ with M_2k(A) via the map

$$ \phi \mapsto (u_{1}^{-1}, \ldots , u_{k}^{-1}, v_{1}, \ldots , v_{k}; \phi |_{A}). $$

Conjugation by an element of W_k acts on this group by a signed permutation of the k pairs $\{u_{i}^{-1},v_{i}\}$, so W_k is a subgroup of the group Sym(2k+1) defined in Lemma 3.1.

An element $\phi \in \widetilde{\mathrm{{Stab}}}^{0}(T)_{A}$ fixes the edge between 1A and x_jA if and only if ϕ(x_j)A=x_jA (equivalently, f_ϕ fixes this terminal vertex). As ϕ(x_j)A=u_jx_jA, this happens if and only if u_j=1 and gives our description of $\widetilde{\mathrm{{Stab}}}^{0}(T)_{e_{j}}$. □

Following Proposition 2.8, each edge stabilizer $\widetilde{\mathrm{{Stab}}}^{0}(T)_{e}$ is an automorphic lift of Stab⁰(T)<Out(F_N) to Aut(F_N).

4.3 Arc stabilizers in the Bass–Serre tree of a collapsed rose with N−2 petals

In the proof of the Theorem A, we will need to understand arc stabilizers for the action of $\widetilde{\mathrm{{Stab}}}(T)$ on the Bass–Serre tree of a collapsed rose with N−2 petals (with the notation of the previous section). In this case, each vertex stabilizer A is isomorphic to F₂, so that $\widetilde{\mathrm{{Stab}}}^{0}(T)_{A} \cong M_{2N-4}(F_{2})$. Proposition 4.1 and Remark 3.7 together imply that the product rank of a vertex stabilizer of T (in Aut(F_N)) is 2N−3, and the product rank of an edge stabilizer is 2N−4. For later arguments, we need to show that the product rank drops further when one takes stabilizers of longer arcs in T.

Lemma 4.2

Let T be a collapsed rose with N−2 petals, and as above let $\widetilde{\mathrm{{Stab}}}(T)$ be the preimage of Stab(T) in Aut(F_N). If α is any edge path of length ≥2 in T then the pointwise stabilizer of α with respect to the $\widetilde{\mathrm{{Stab}}}(T)$–action on T has product rank at most 2N−5.

Proof

It is enough to prove the result for a path α={e,e′} of length two, and as product rank is preserved under passing to finite-index subgroups, we may pass to $G:=\widetilde{\mathrm{{Stab}}}^{0}(T)$. We want to show that the intersection G_e∩G_e′ of the edge stabilizers in G has product rank at most 2N−5. As above, suppose that A≅F₂ is the subgroup of F_N which stabilizes the vertex adjacent to e and e′. Without loss of generality we may assume that e is the edge between vertices with stabilizers A and $x_{1}Ax_{1}^{-1}$ respectively. Then G_e is the subgroup of Aut(F_N) that preserves A, sends x₁ to x₁v₁, and sends each x_i to a word of the form u_ix_iv_i for 2≤i≤N−2 (where as above each u_i,v_i∈A). Note that we can replace a stable letter x_i with $ux_{i}^{\pm 1}v$ if 2≤i≤N−2 and u,v∈A. We may also replace x₁ with x₁v for some v∈A without changing the description of G_e. Up to the above replacements, there are three possibilities for the stabilizer of the second vertex of e′: it is either of the form $x_{j}Ax_{j}^{-1}$ for some j satisfying 2≤j≤N−2, of the form $x_{1}^{-1}Ax_{1}$, or of the form (wx₁)A(wx₁)⁻¹ for some nontrivial w∈A. In the first case, we see that G_e∩G_e′ consists of automorphisms ϕ preserving A with the added restriction that both u_j=1 and u₁=1 in the above notation. Hence the intersection decomposes as an exact sequence

$$ 1 \to A^{2k-2} \to G_{e} \cap G_{e'} \to {\mathrm{{Aut}}}(A) \to 1, $$

where k=N−2. The product rank of the kernel is then 2N−6 and the product rank of Aut(A)=Aut(F₂) is one, so the product rank of the intersection is at most 2N−5.

In the second case, where the terminal vertex of e′ has stabilizer $x_{1}^{-1}Ax_{1}$, a similar argument applies. One sees that G_e∩G_e′ is the subgroup of the stabilizer of v_A which fixes x₁ (hence u₁=v₁=1), and we have the same exact sequence as above.

In the final case, the intersection G_e∩G_e′ is given by the automorphisms in G_e which also fix the subgroup (wx₁)A(wx₁)⁻¹. Note that

$$ \phi ((wx_{1})A(wx_{1})^{-1})=\phi (w)\phi (x_{1}Ax_{1}^{-1})\phi (w)^{-1}= \phi (w)x_{1}Ax_{1}^{-1}\phi (w)^{-1} $$

as elements of G_e preserve $x_{1}Ax_{1}^{-1}$. If ϕ is also an element of G_e′ then

$$ (wx_{1})A(wx_{1})^{-1}= \phi (w)x_{1}Ax_{1}^{-1}\phi (w)^{-1}. $$

This implies that ϕ(w)=w as w∈A. Therefore in the exact sequence

$$ 1 \to A^{2N-5} \to G_{e} \to {\mathrm{{Aut}}}(A) \to 1, $$

the intersection G_e∩G_e′ projects to a subgroup of Aut(A) fixing the element w. However, parts 1 and 3 of Lemma 3.11 imply that a subgroup of G_e≅A^2N−5⋊Aut(A) of maximal product rank projects to a subgroup of Aut(A) containing a nonabelian subgroup of inner automorphisms, and therefore cannot fix any w∈A. It follows that G_e∩G_e′ has product rank at most 2N−5. □

Proposition 4.3

Let T be a collapsed rose with N−2 petals, and let D be a subgroup of $\widetilde{\mathrm{{Stab}}}(T)< {\mathrm{{Aut}}}(F_{N})$ generated by 2N−3 pairwise-commuting full-sized subgroups. Then D has a unique global fixed point v in T. The normalizer of D in $\widetilde{\mathrm{{Stab}}}(T)$ also fixes v.

Proof

If a fixed vertex of D exists then it is unique as stabilizers of edges in T have product rank 2N−4. By uniqueness, such a fixed point will also be invariant under the normalizer of D in $\widetilde{\mathrm{{Stab}}}(T)$. We are left with the matter of proving such a fixed point exists.

Suppose D is generated by D₁,D₂,…,D_2N−3. Firstly, suppose that some subgroup, for instance D_2N−3, contains a hyperbolic element g with respect to the action on T. As each subgroup D_i for i<2N−3 commutes with g, the subgroup 〈D₁,D₂,…,D_2N−4〉 acts on the axis A_g of g preserving the orientation. The commutator subgroup $D_{i}'$ of each D_i acts trivially on this axis, and as these groups are full-sized we obtain a direct product of 2N−4 nonabelian free groups fixing a line in T. This contradicts Lemma 4.2. We may therefore assume that each subgroup D_i consists of elliptic elements. Any product of commuting elliptic elements is also elliptic, so every element of D is elliptic. If D does not have a global fixed point, then as in [Ser80, I.6.5, Exercise 2] the group D is not finitely generated and fixes an end of T. Every element of D fixes a half-line towards this end, so that we can find a finitely generated subgroup $\langle D_{1}',D_{2}',\ldots , D_{2N-3}'\rangle$ with each $D_{i}'$ a nonabelian free group fixing a half-line in T, which again contradicts Lemma 4.2. Hence D has a global fixed point in T. □

4.4 Stabilizers of cages

A splitting T of F_N is a cage if T is a free splitting and the quotient graph T/F_N is isomorphic to a cage (a connected graph with two vertices and no loop edges). Suppose that T/F_N is a cage with k edges. Pick adjacent vertices v and w in T with stabilizers A and B. Let e be the edge from v to w and let e₁,…,e_k−1 be edges based at v representing the other k−1 edge orbits in T. If x_i is an element that takes the terminal vertex of e_i to w, then F_N is generated by A, B, and x₁,…,x_k−1.

As above, take G=Stab⁰(T) and $\tilde{G}$ its preimage in Aut(F_N). The stabilizer $\tilde{G}_{e}$ of e gives an automorphic lift of G. As in the proof of Proposition 4.1, every element Φ∈Stab⁰(T) has a unique representative $\phi \in \tilde{G}_{e}$ such that

• ϕ restricts to an automorphism of A and B.

• For each x_i we have ϕ(x_i)=a_ix_ib_i for some a_i∈A and b_i∈B.

It follows that Stab⁰(T) fits in the exact sequence

$$ 1\to A^{k-1} \times B^{k-1} \to {\mathrm{Stab}}^{0}(T) \to {\mathrm{{Aut}}}(A) \times {\mathrm{{Aut}}}(B) \to 1. $$

Furthermore, this sequence splits so that Stab⁰(T) is a semidirect product of Aut(A)×Aut(B) with A^k−1×B^k−1, where Aut(A) acts diagonally on A^k−1 and trivially on B^k−1, and the action of Aut(B) on A^k−1×B^k−1 acts trivially on A^k−1 and diagonally on B^k−1. In the proof of Proposition 5.1, we will use this decomposition to calculate rk_F(Stab⁰(T)).

5 Fixed splittings of subgroups with maximal product rank

In this section, we prove the following proposition:

Proposition 5.1

Suppose that $G < \mathrm{{\mathrm{IA}}_{N}(\mathbb{Z}}/\mathrm{3\mathbb{Z})}$ and rk_F(G)=2N−4. Any splitting fixed by G has one F_N-orbit of vertices (i.e. is a collapsed rose). Any two free splittings fixed by G are compatible.

Proposition 5.1 follows from two lemmas that will be proved below. Lemma 5.4 asserts that if a subgroup $G < \mathrm{{\mathrm{IA}}_{N}(\mathbb{Z}}/\mathrm{3\mathbb{Z})}$ fixes two incompatible free splittings, then G fixes a free splitting with at least two F_N-orbits of vertices. Lemma 5.5 then shows that such a splitting cannot be fixed by a group with maximal product rank (in other words, subgroups with maximal product rank can only fix collapsed roses). We delay the full proof for the time being to first state two consequences of this proposition.

We say that a free splitting T is G-unrefinable if it admits no G-invariant refinement that is a free splitting. We have the following useful corollary:

Corollary 5.2

Suppose $G < \mathrm{{\mathrm{IA}}_{N}(\mathbb{Z}}/\mathrm{3\mathbb{Z})}$ and rk_F(G)=2N−4.

• The group G fixes a unique G-unrefinable free splitting. If this splitting is nontrivial then it is a collapsed rose.

• If T is the unique G-unrefinable collapsed rose fixed by G, then T is also fixed by the normalizer $N_{{\mathrm{{Out}}}(F_{N})}(G)$ of G in Out(F_N).

Proof

Take X to be the set of all one-edge free splittings preserved by G. These are all compatible by Proposition 5.1, so X is finite and there is a common refinement T collapsing to every element of X ([SS03, Theorem 5.16] or [GL17, Proposition A.17]). Also by Proposition 5.1, the tree T is a collapsed rose. Any other G-invariant free splitting is a common refinement of a subset of X (this follows from the fact that G is contained in $\mathrm{{\mathrm{IA}}_{N}(\mathbb{Z}}/\mathrm{3\mathbb{Z})}$ and Proposition 2.5: if G fixes a free splitting T′ then G also fixes all of the one-edge splittings to which T′ collapses). It follows that T is the unique G-unrefinable, G-invariant free splitting. By uniqueness, T is invariant under the normalizer of G in Out(F_N). □

Corollary 5.3

Let $G < \mathrm{{\mathrm{IA}}_{N}(\mathbb{Z}}/\mathrm{3\mathbb{Z})}$ be such that rk_F(G)=2N−4. If G fixes the Bass–Serre tree T of a collapsed rose with N−2 petals, then this rose is unique and the normalizer N(G) of G in Out(F_N) also fixes T.

Proof

A G-invariant refinement of such a rose would have to have N−1 or N petals. However it follows from Proposition 4.1 that the stabilizer of a collapsed rose with N−1 petals is virtually abelian, while a rose with N petals has finite stabilizer. Hence if G fixes a collapsed rose with N−2 petals then this rose is G-unrefinable and invariant under the normalizer of G in Out(F_N). □

We move on to the required lemmas.

Lemma 5.4

If $G < \mathrm{{\mathrm{IA}}_{N}(\mathbb{Z}}/\mathrm{3\mathbb{Z})}$ fixes incompatible free splittings T and T′ then G fixes a free splitting which contains at least two orbits of vertices.

Proof

As refinements of incompatible trees are also incompatible, we may assume that the two splittings T and T′ are incompatible and G-unrefinable. Let $\tilde{G}$ be the preimage of G in Aut(F_N). By Proposition 6.2 of [GH21], any two G-unrefinable free splittings belong to the same deformation space when viewed as $\tilde{G}$-trees (equivalently, T and T′ have the same set of elliptic subgroups with respect to the $\tilde{G}$-action). However, if there are incompatible trees in a deformation space then there must be trees with at least two orbits of vertices. We give a brief explanation via folding. As T and T′ belong to the same deformation space, there exists a $\tilde{G}$ equivariant map f:T→T′ taking edges in T to (possibly trivial) edge paths in T′ (see, e.g., [GL07]). The map f decomposes as a collapse f₀:T→T₀ followed by a morphism f₁:T₀→T₁ that does not collapse edges. The map f₁ is nontrivial as T and T′ are incompatible, which implies there exist edges e≠e′ based at the same vertex in T₀ such that the edge paths f(e) and f(e′) have a common initial egde (if f₁ were locally injective, then f₁ would be an isomorphism). Partially folding e and e′ along a small initial segment gives a new $\tilde{G}$ tree T₁ with an extra $\tilde{G}$-orbit of vertices at the fold. There is an induced morphism f₂:T₁→T′, which implies that any edge stabilizer of T₁ fixes a nondegenerate arc in T′. Thus $F_{N} < \tilde{G}$ has no nontrivial edge stabilizer in T₁, and T₁ is a G-invariant free splitting with at least two orbits of edges. □

We now show that the stabilizer of any splitting with more than one orbit of vertices does not have maximal product rank. Proposition 5.1 follows immediately.

Lemma 5.5

Let T be a splitting of F_N which contains at least two F_N-orbits of vertices. Then ${{\mathrm{rk}_{F}}}({\mathrm{{Stab}}}_{{\mathrm{{Out}}}(F_{N})}(T)) \leq 2N-5$.

Proof

Suppose for a contradiction that rk_F(Stab(T))=2N−4. Then the intersection of Stab(T) with the finite index subgroup $\mathrm{{\mathrm{IA}}_{N}(\mathbb{Z}}/\mathrm{3\mathbb{Z})< {\mathrm{{Out}}}(F_{N})}$ also contains a direct product D of 2N−4 nonabelian free groups, and D preserves every collapse of T (Proposition 2.5). We may therefore replace T with a collapse T′ that has exactly two orbits of vertices v and w. Collapsing all loop edges in the quotient graph then gives us a cage S with k≥1 edges (allowing the degenerate case of a one-edge separating splitting). As $D < \mathrm{{\mathrm{IA}}_{N}(\mathbb{Z}}/\mathrm{3\mathbb{Z})}$, we have D<Stab⁰(S). If A and B are free factor representatives of the vertex stabilizers in S then, as in Sect. 4.4, the group Stab⁰(S) decomposes as a short exact sequence

$$ 1\to A^{k-1} \times B^{k-1} \to {\mathrm{Stab}}^{0}(S) \to {\mathrm{{Aut}}}(A) \times {\mathrm{{Aut}}}(B) \to 1. $$

Suppose A and B are of rank n and m respectively, so that N=n+m+(k−1). If A and B are both noncyclic, then

$$ {{\mathrm{rk}_{F}}}({\mathrm{{Aut}}}(A))+{{\mathrm{rk}_{F}}}({\mathrm{{Aut}}}(B))= (2n-3) + (2m-3) =2N-2k-4 $$

and

$$ {{\mathrm{rk}_{F}}}(A^{k-1} \times B^{k-1}) = 2k-2. $$

Hence rk_F(Stab⁰(S))≤(2N−2k−4)+2k−2=2N−6. If both A and B are cyclic (possibly trivial), then Stab⁰(S) is virtually abelian, so rk_F(Stab⁰(S))=0. Suppose A is nonabelian and B is cyclic. Then rk_F(A^k−1×B^k−1)=k−1. If B is trivial then A is rank N−k+1 and the vertex w has valence at least 3, so that k≥3. Then rk_F(Aut(A)×Aut(B))=rk_F(Aut(A))=2(N−k+1)−3. Hence

$$ {{\mathrm{rk}_{F}}}({\mathrm{Stab}}^{0}(S)) \leq 2(N-k+1)-3 + k-1 = 2N -2 -k \leq 2N-5. $$

Similarly, if B is infinite cyclic then A is rank N−k and the same computation gives

$$ {{\mathrm{rk}_{F}}}({\mathrm{Stab}}^{0}(S)) \leq 2N-4-k \leq 2N-5. $$

 □

6 Completing the proofs of theorems A and B

We are now armed with enough knowledge to complete the proof of Theorem A.

Theorem 6.1

Theorem A

Let N≥3 and suppose

$$ D=\langle D_{1}, D_{2} \ldots , D_{2N-4}\rangle < {\mathrm{{Out}}}(F_{N}) $$

is generated by 2N−4 pairwise-commuting full-sized subgroups D₁,…,D_2N−4. Then there is a unique collapsed rose with N−2 petals that is fixed by D.

Remark 6.2

Once we have established the existence of the collapsed rose in this theorem, its uniqueness is assured by Corollary 5.3, from which it follows that the collapsed rose is also fixed by the normalizer of D. This observation about normalizers will be useful in the inductive proof that follows.

Proof

We first reduce to the case where $D < \mathrm{{\mathrm{IA}}_{N}(\mathbb{Z}}/\mathrm{3\mathbb{Z})}$. Let $D_{i}'=D_{i} \cap \mathrm{{\mathrm{IA}}_{N}(\mathbb{Z}}/\mathrm{3\mathbb{Z})}$, note that $D_{i}'$ is full-sized and let

$$ D' = \langle D_{1}', D_{2}',\ldots , D_{2N-4}'\rangle . $$

As D′ is normal in D, if D′ fixes a collapsed rose with N−2 petals, then so does D, by Remark 6.2. Thus we may assume without loss of generality that $D < \mathrm{{\mathrm{IA}}_{N}(\mathbb{Z}}/\mathrm{3\mathbb{Z})}$.

When N=3, a rose with N−2 petals is simply a nonseparating free splitting. By Theorem 2.6, a group D=〈D₁,D₂〉 in $\mathrm{{\mathrm{IA}}_{3}(\mathbb{Z}}/\mathrm{3\mathbb{Z})}$ fixes such a splitting.

For the inductive step, we will take the preimage $\tilde{D}$ of D in Aut(F_N) and build an action of $\tilde{D}$ on a tree T such that the restriction to the inner automorphisms F_N<Aut(F_N) is a free splitting with N−2 F_N-orbits of edges. This splitting is then D-invariant (Lemma 2.7) and is necessarily an N−2 rose by Corollary 5.2. We build the tree as a graph of actions (see [Lev94]). A graph of actions for a group H consists of the following data:

• A graph of groups decomposition of H.

• For every vertex u in the graph of groups, a tree T_u equipped with an action of the vertex group H_u.

• For every oriented edge e with terminus u, a fixed point x_e of H_e<H_u in T_u.

Given a graph of actions where the vertex trees are all simplicial, if S is the Bass–Serre tree of the underlying graph of groups then one obtains a refinement T of S from the graph of actions. There is a natural collapse map T→S, where the preimage of a vertex $\tilde{u} \in S$ is a copy of T_u. The points x_e are used as gluing instructions for the endpoints of the edges of S into the trees T_u. We will take S to be any one-edge nonseparating free splitting invariant under D, which exists by Theorem 2.6. Let A≅F_N−1 be a free factor whose conjugates form the vertex stabilizers of S, and let b_A (hereafter shortened to b) be the vertex fixed by A. Let D_A be the image of D in Out(A). By considering product rank in the short exact sequence

$$ 1 \to A \times A \to {\mathrm{{Stab}}}^{0}(S) \to {\mathrm{{Out}}}(A) \to 1, $$

the kernel K of the homomorphism D→D_A has rk_F(K)≤2. By Proposition 2.3, after reordering the D_i, the groups D₃/K,…,D_2N−4/K are full-sized in D_A. In particular, D_A is in the normalizer of the group generated by a commuting family of 2N−6 full-sized groups in Out(A). By induction and Remark 6.2, we know that D_A fixes a collapsed rose T_A with N−3 petals. Let $\tilde{D}_{A}$ be the preimage of D_A in Aut(A). As $\tilde{D}_{b}$ preserves A, the vertex group $\tilde{D}_{b}$ acts on T_A via the projection $\tilde{D}_{b} \to {\mathrm{{Aut}}}(A)$. In order to show that we can define a graph of actions, we need to take an edge e adjacent to b and check that $\tilde{D}_{e}$ has a fixed point with respect to its action on T_A. We let $f: \tilde{D}_{e} \to \tilde{D}_{A}$ be the map factoring through $\tilde{D}_{b}$ that determines this action. We have the following commutative diagram.

The map from $\tilde{D}_{e}$ to D is an isomorphism as S is a free splitting, so that $\tilde{D}_{e}$ is an automorphic lift as described in Sect. 2.4. Without loss of generality, we can take e corresponding to the stable letter x₁, so that the stabilizers of its endpoints are A and $x_{1}A x_{1}^{-1}$. As in Sect. 4.2, every automorphism $\phi \in \tilde{D}_{e}$ restricts to an automorphism ϕ_A of A and satisfies ϕ(x₁)=x₁a for some a∈A. It follows that the kernel of f is a free group generated by these right transvections. Applying Proposition 2.3 to $\tilde{D}_{e} \cong D$, after permuting the D_i, the images of D₂,…,D_2N−4 are full-sized in Im(f). Therefore Im(f) is contained in the normalizer of a commuting family of 2N−5=2(N−1)−3 full-sized groups in Aut(A), so by Proposition 4.3, the group Im(f) has a fixed point x_e in T_A.

It follows that $\tilde{D}$ admits a graph of actions with the Bass–Serre tree S and vertex tree T_A, so that the refinement of S determined by this graph of actions is a free splitting of F_N with N−2 orbits of edges. □

Theorem 6.3

Theorem B

Let N≥3 and suppose D<Aut(F_N) is generated by a commuting family of 2N−3 full-sized subgroups. Then, the image of D in Out(F_N) fixes a unique collapsed rose with N−2 petals, and D acts on the Bass–Serre tree of this collapsed rose with a unique global fixed point.

Proof

The image $\bar{D}$ of D in Out(F_N) is contained in the normalizer of a commuting family of 2N−4 nonabelian free groups (the kernel of the map $D \to \bar{D}$ is free and normal, so we may apply Proposition 2.3 with l=1). Hence $\bar{D}$ fixes a collapsed rose T with N−2 petals in the boundary of Outer space, and D acts on this tree. Proposition 4.3 then states that D has a unique global fixed point with respect to the action on T. □

7 Algebraic descriptions of the direct products and their centralizers

In this section we prove Theorems C, D and F from the introduction, where the relevant notation was established. We prove an Out(F_N)-version of Theorem C in Theorem 7.2.

Theorem 7.1

Theorem C

Let N≥3 and let D<Aut(F_N) be a direct product of 2N−3 nonabelian free groups. Then a conjugate of D is contained in one of the following subgroups.

• L₁×⋯×L_N−2×R₁×⋯×R_N−2×I(A)

• $L_{1}^{\tau }\times \cdots \times L_{N-2}^{\tau }\times R_{1}^{\tau } \times \cdots \times R_{N-2}^{\tau }\times I(A)^{\tau }\times \langle \tau \rangle$

• $\langle L_{1}, \tau \rangle \times L_{2}^{\tau }\times \cdots \times L_{N-2}^{\tau }\times R_{1}^{\tau }\times \cdots \times R_{N-2}^{ \tau }\times I(A)^{\tau}$

• $L_{1}^{\tau }\times \cdots \times L_{N-2}^{\tau }\times R_{1}^{\tau } \times \cdots \times R_{N-2}^{\tau }\times \langle I(A), \tau \rangle$

In the second case, if D=D₁×⋯×D_2N−3, then each summand D_i is contained in a unique subgroup of the form $\langle L_{i}^{\tau}, \tau \rangle$, $\langle R_{i}^{\tau}, \tau \rangle$, or 〈I(A)^τ,τ〉. In the remaining cases, the inclusion respects the visible direct product decompositions, up to permuting the summands.

Proof

By Theorem B, there exists an action of D on the Bass–Serre tree T of a collapsed rose with N−2 petals such that D has a (unique) global fixed point. Up to conjugation, we may assume that the collapsed rose has vertex group A=〈a₁,a₂〉 and stable letters x₁,…,x_N−2, and $D < \widetilde{\mathrm{{Stab}}}(T)_{A}$ (i.e. the vertex fixed by D in T has F_N–stabilizer equal to A). By Proposition 4.1, we have

$$ \widetilde{\mathrm{{Stab}}}(T)_{A}=M_{2N-4}(A) \rtimes W_{N-2}, $$

where W_N−2 acts as a subgroup of the group Sym(2N−3) of automorphisms of M_2N−4(A) defined in Lemma 3.1. By Proposition 3.12, the projection of D to W_N−2 is trivial, so that D is contained in $\widetilde{\mathrm{{Stab}}}^{0}(T)_{A}=M_{2N-4}(A)$. Under the isomorphism between $\widetilde{\mathrm{{Stab}}}^{0}(T)_{A}$ and M_2N−4(A) given in Proposition 4.1, the groups L_i and R_i are taken to factors in the A^2N−4 subgroup of M_2N−4(A), the inner automorphisms by elements of A are taken to J, and τ is taken to $\underline{\tau}$ (using the notation of Sect. 3). The possibilities for D are then given by Theorem 3.8. □

Theorem D from the introduction is proved in exactly the same way as above and the proof is omitted. In the following theorem we blur the distinction between L_i and R_i and their (isomorphic) images in Out(F_N).

Theorem 7.2

Let N≥3 and let D<Out(F_N) be a direct product of 2N−4 nonabelian free groups. Then a conjugate of D is contained in one of the following subgroups.

• L₁×⋯×L_N−2×R₁×⋯×R_N−2

• $L_{1}^{\tau }\times \cdots \times L_{N-2}^{\tau }\times R_{1}^{\tau } \times \cdots \times R_{N-2}^{\tau }\times \langle [\tau ] \rangle$

• $\langle [\tau ], L_{1} \rangle \times L_{2}^{\tau }\times \cdots \times L_{N-2}^{\tau }\times R_{1}^{\tau }\times \cdots \times R_{N-2}^{ \tau}$

In the second case, if D=D₁×⋯×D_2N−4, then each summand D_i is contained in a unique subgroup of the form $\langle L_{i}^{\tau}, [\tau ] \rangle$ or $\langle R_{i}^{\tau}, [\tau ]\rangle$. In the remaining cases, the inclusion respects the visible direct product decompositions, up to permuting the summands.

Proof

We apply Theorem A to find a D-invariant collapsed rose with N−2 petals in the boundary of Outer space. We then conjugate D so that this rose T is the one given by A and x₁,…,x_N−2. In this case it is more natural to see Stab⁰(T)≅M_2N−5(A) decomposing as the exact sequence:

$$ 1 \to L_{1} \times \cdots \times L_{N-2} \times R_{1} \times \cdots \times R_{N-2} \to {\mathrm{Stab}}^{0}(T) \to {\mathrm{{Out}}}(A) \to 1, $$

so that Stab(T)=Stab⁰(T)⋊W_N−2 and W_N−2 acts by signed permutations of the (R_i,L_i) pairs of factors in the kernel of this exact sequence. Automorphic lifting (see the proof of Proposition 2.8) tells us that ${\mathrm{{Stab}}}^{0}(T)\cong \widetilde{\mathrm{{Stab}}}^{0}(T)_{e}$ for any edge e in the tree. If e is the edge between the vertices corresponding to the cosets 1A and x₁A in the Bass–Serre tree, then in Proposition 4.1 we have seen that each $\phi \in \widetilde{\mathrm{{Stab}}}^{0}(T)_{e}$ preserves A, maps x₁↦x₁v₁ and for 2≤i≤N−2 maps x_i↦u_ix_iv_i with the u_i,v_i∈A. This gives an isomorphism between ${\mathrm{{Stab}}}^{0}(T)= \widetilde{\mathrm{{Stab}}}^{0}(T)_{e}$ and M_2N−5(A) via

$$ \Theta (\phi ) = (u_{2}^{-1},\ldots , u_{N-2}^{-1},v_{1}, \ldots v_{N-2} ; \phi |_{A} ). $$

Although the images of L₂,…,L_N−2 and R₁,…,R_N−2 are easy to see under Θ, the ‘natural’ representatives of the elements of L₁ are not in $\widetilde{\mathrm{{Stab}}}^{0}(T)_{e}$. However the transvection ϕ mapping x₁↦ax₁ is equivalent in Out(F_N) to the automorphism ϕ′ sending x₁ to x₁a and conjugating every other basis element by a⁻¹, so that $\Theta (\phi ')=(a, \ldots , a; {\mathrm{{ad}}}_{a}^{-1})$. It follows that L₁ is mapped to the subgroup J of M_2N−5(A) (using the notation of Sect. 3). With some care one can then check that, as in the Aut proof, W_N−2 acts on M_2N−5(A) through Θ as a subgroup of the group Sym(2N−4) defined in Sect. 3. In particular, Proposition 3.12 then tells us that the projection of D to W_N−2 is trivial and D<Stab⁰(T). The result then follows by combining Theorem 3.8 with the above isomorphism and the observation that Θ(L₁)=J. □

Proposition 7.3

Let N≥3 and let D<Aut(F_N) be a direct product of 2N−3 nonabelian free groups in Aut(F_N). Then the centralizer of D is either trivial or generated by a Nielsen automorphism.

Proof

Let C=C(D) be the centralizer of D, and let $\overline{C}$, $\overline{D}$ be the respective images of C and D in Out(F_N). As in the proof of Theorem B, the group $\overline{D}$ is contained in the normalizer of a direct product of 2N−4 nonabelian free groups, so by Theorem A it fixes a unique collapsed rose with N−2 petals in the boundary of Outer space. By uniqueness $\overline{C}$ also fixes this rose, so C acts on the associated Bass–Serre tree T. The group D acts on T with a unique fixed point (Proposition 4.3), therefore C also fixes this point. Hence C is in the subgroup H=M_2N−4(A)⋊W_N−2 described in Proposition 4.1. The second part of Proposition 3.12 tells us that the projections of C and D to the W_N−2<Sym(2N−3) factor are trivial, so C<M_2N−4(A). We can therefore apply Part 2 of Theorem 3.8: if the centralizer of D in M_2N−4(A) is nontrivial it is generated by the element ${\underline{\tau }}$, and this is mapped to a Nielsen transformation τ under the isomorphism between the group M_2N−4(A) and the point stabilizer in T given in Proposition 4.1. □

Corollary 7.4

Theorem F

Let N≥3 and suppose Γ is a finite-index subgroup of Aut(F_N). If f:Γ→Aut(F_N) is an injective homomorphism then every power of a Nielsen automorphism is mapped to a power of a Nielsen automorphism under f.

Proof

If ϕ is the power of a Nielsen automorphism then ϕ centralizes a direct product D<Aut(F_N) of 2N−4 nonabelian free groups (see the second case of Theorem C). By taking finite-index subgroups of the factors, we can assume that D<Γ, so that f(ϕ) centralizes f(D). Hence f(ϕ) is also a power of a Nielsen automorphism by Proposition 7.3. □

8 Ascending chains of direct products

As well as taking direct products of free groups in Aut(F_N) or Out(F_N) with a maximal number of direct factors, one might also ask about maximality with respect to inclusion. For arbitrary groups, one must be very careful: if G is a group with product rank k and $\mathcal{D}$ is the poset of direct products of k nonabelian free groups in G (ordered by containment), it is not generally true that $\mathcal{D}$ contains maximal elements. One reason for this is the existence of locally free groups that are not free.

Recall that a group G is called locally free if and only if every finitely generated subgroup of G is free. If G is countable, this is equivalent to the condition that G is the direct limit of its free subgroups. The simplest example of a locally free group that is not free is $\mathbb{Q}$. The free product $\mathbb{Q} \ast \mathbb{Q}$ is also locally free and clearly contains nonabelian free groups. However, divisibility is not the only reason why a locally free group can fail to be free. We are grateful to Henry Wilton for directing us to the following example of Kurosh: take the one-relator group G=〈a,b,t | t[a,b]t⁻¹=a〉, and let H be the kernel of the map to $\mathbb{Z}$ given by a,b↦0, t↦1. The group G is the fundamental group of the space X_G obtained from a one-holed torus T by gluing the boundary curve of T to a simple closed curve on T. The group H is then the fundamental group of an infinite chain of surfaces (shown in Fig. 2).

Figure 2

Kurosh’s example of a locally free group that is not free is the fundamental group of the above infinite chain of tori.

The group H has an infinite presentation with generating set $\{a_{i},b_{i}: i \in \mathbb{Z}\}$ and relations [a_i,b_i]=a_i+1 for all $i \in \mathbb{Z}$. The subgroup generated by a_−n,b_−n,…,a_n,b_n is freely generated by a_−n and b_−n,b_−n+1,…,b_n, which implies that H is locally free. However, H is not free as H is not residually nilpotent: the relations let us write each a_i as an arbitrarily long iterated commutator, so a_i is trivial in every nilpotent quotient of H.

The group G containing H is well-behaved—it is hyperbolic and the fundamental group of a 3-manifold with boundary (this follows by constructing X_G in $\mathbb{R}^{3}$ and thickening). Following the classification results in Sect. 7, we would like to rule out this behaviour in the mapping torus $M_{\tau }= F_{2} \rtimes _{\tau }\mathbb{Z}$, where we will take F₂=〈a,b〉 and τ the automorphism taking a↦ab and fixing b. In the spirit of the rest of the paper, we embed M_τ in Aut(F₂) by identifying F₂⋊1 with the inner automorphisms

Proposition 8.1

A subgroup of M_τ is free if and only if it does not contain a subgroup isomorphic to $\mathbb{Z}^{2}$. Any free subgroup of M_τ is contained in a maximal one.

Proof

The second statement follows from the first via Zorn’s Lemma. In more detail, if we have an ascending chain H₁<H₂<H₃<⋯ of free subgroups of M_τ, then the union H=∪H_i does not contain a subgroup isomorphic to $\mathbb{Z}^{2}$ (as it would be contained in one of the H_i). Therefore H is free, and we can apply Zorn’s Lemma.

We are left with the trickier task of proving the first assertion. To do this, we look at the limiting tree T of the automorphism τ in the boundary of Outer space (see [CL95]). This is a cyclic splitting with vertex stabilizers conjugate to Fix(τ)=〈aba⁻¹,b〉 and edge stabilizers conjugate to 〈b〉. As T is invariant under [τ], there is an action of M_τ on T with vertex stabilizers conjugate to 〈aba⁻¹,b〉×〈τ〉 and edge stabilizers conjugate to $\mathbb{Z}^{2}=\langle b, \tau \rangle$. At a vertex v, there are two Stab(v)–orbits of adjacent edges. If Stab(v)=〈aba⁻¹,b〉, these are the conjugacy classes of 〈aba⁻¹,τ〉 and 〈b,τ〉 in Stab(v).

Recall that a cylinder in T is a subtree C_g that is fixed pointwise by a nontrivial element g of M_τ. In F₂ edge stabilizers are malnormal, so if a cylinder C_g contains more than one edge then g∉F₂. Hence g=ad_xτ^k for some x∈F₂ and k≠0. The subgroup of F₂ fixed by g is nonabelian, as g commutes with any inner automorphism fixing an edge in its cylinder. Hence g is similar to τ^k by Collins–Turner (Proposition 3.3). Hence the cylinder is a star of radius one (if τ fixes an edge then the edge is adjacent to the vertex with stabilizer Fix(τ)). This shows that every cylinder in T is either a point, a single edge, or a star of radius one.

Let H be a subgroup of M_τ that does not contain $\mathbb{Z}^{2}$. Then for every vertex v∈T, the stabilizer H_v is free. As above we may assume that Stab(v)=〈aba⁻¹,b〉×〈τ〉. The ‘exceptional’ case is where H_v∩〈τ〉 is nontrivial. Then H_v<〈τ〉 as H contains no $\mathbb{Z}^{2}$ subgroups. Then the H–stabilizer of every edge adjacent to v is also equal to H_v. The ‘generic’ case is where H_v∩〈τ〉 is trivial. Then H_v embeds into 〈aba⁻¹,b〉 via the projection to this factor, and under this projection each adjacent edge stabilizer is contained in a Stab(v)–conjugate of 〈aba⁻¹〉 or 〈b〉. Hence H_v splits relative to its adjacent edge groups via the free splitting S_v:=〈aba⁻¹〉∗〈b〉.

One can therefore blow up each generic vertex group via the splitting S_v. In the new H–tree T′, the nontrivial edge stabilizers are equal to their adjacent vertex stabilizers, so the tree is a union of cylinders with identical cyclic edge and vertex groups, necessarily separated by edges with trivial stabilizers. As all new edges in the blow-up have trivial stabilizers, each cylinder in T′ is unchanged from T and is either a point, a single edge, or a star of radius one. It follows that any setwise stabilizer of a cylinder fixes a point in that cylinder, and therefore fixes the cylinder pointwise (as the edge and vertex groups in cylinders are all identical). This means we can collapse each cylinder to a point, giving a tree T″ on which H acts with trivial edge stabilizers and vertex stabilizers that are either $\mathbb{Z}$ or trivial. Hence H is free. □

Corollary 8.2

Corollary E

Let N≥3 and let $\mathcal{D}$ be the family of subgroups of either Aut(F_N) or Out(F_N) that are direct products of 2N−3 or 2N−4 nonabelian free groups, respectively. Then every $D \in \mathcal{D}$ is contained in a maximal element (with respect to inclusion).

Proof

We limit the proof to the Aut(F_N) case. Let $D \in \mathcal{D}$. We say a direct summand D_i of D is twisted if it acts nontrivially on the homology of A=〈a₁,a₂〉, and is untwisted otherwise (roughly speaking, this detects if τ is seen in D_i). If at least two summands are twisted, then the proof of Theorem 3.8 implies that each summand of D is contained in a unique $F_{2} \times \mathbb{Z}$ given by $\langle L_{i}^{\tau}, \tau \rangle$, $\langle R_{i}^{\tau}, \tau \rangle$, or 〈I(A)^τ,τ〉, and any $D' \in \mathcal{D}$ containing D also satisfies this conclusion. One then chooses maximal free subgroups of each $F_{2} \times \mathbb{Z}$ containing each D_i. Otherwise, we are in one of the remaining cases of Theorem C, where each summand is embedded in an F₂ or a subgroup isomorphic to M_τ, and this enveloping group is a maximal direct product of full-sized subgroups of Aut(F_N) by Theorem D. Hence taking each F₂ and a maximal free subgroup of the M_τ summand (which exists by Proposition 8.1) will suffice. □