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.
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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. NielsenThurston theory [Thu88] provides a potent geometric description of the individual elements of mapping class groups, and the traintrack 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 NielsenThurston theory, one knows that, up to finiteindex, every abelian subgroup of the mapping class group is generated by combinations of Dehn twists in a collection of disjoint curves and, optionally, a pseudoAnosov 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 pseudoAnosov pieces and compatibility conditions for the curve systems. With these in hand, one can organise the commensurability classes of maximalrank 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 lowgenus exceptions, every isomorphism between finiteindex 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 maximalrank 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 fullsized groups (groups that contain nonabelian 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 fixedpoint 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 F_{2}; Horbez and Wade [HW20] proved that this number is 2N−4 (one less than the cohomological dimension).
We fix a basis {a_{1},a_{2},x_{1},…,x_{N−2}} of F_{N} and consider the direct product D of the 2N−4 copies of F_{2} in Out(F_{N}) obtained by multiplying the elements x_{1},…,x_{N−2} on the left and right by elements of 〈a_{1},a_{2}〉. This group D fixes a graphofgroups decomposition of F_{N} with a single vertex group given by A=〈a_{1},a_{2}〉 and N−2 loops with trivial edge stabilizers (with x_{1},…,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).
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 fullsized 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 fullsized 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 {a_{1},a_{2},x_{1},…,x_{N−2}} of F_{N} and let A be the free factor generated by a_{1} and a_{2}. 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 a_{1}↦a_{1}a_{2} 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.

L_{1}×⋯×L_{N−2}×R_{1}×⋯×R_{N−2}×I(A)

\(L_{1}^{\tau }\times \cdots \times L_{N2}^{\tau }\times R_{1}^{\tau } \times \cdots \times R_{N2}^{\tau }\times I(A)^{\tau }\times \langle \tau \rangle \)

\(\langle L_{1}, \tau \rangle \times L_{2}^{\tau }\times \cdots \times L_{N2}^{\tau }\times R_{1}^{\tau }\times \cdots \times R_{N2}^{ \tau }\times I(A)^{\tau}\)

\(L_{1}^{\tau }\times \cdots \times L_{N2}^{\tau }\times R_{1}^{\tau } \times \cdots \times R_{N2}^{\tau }\times \langle I(A), \tau \rangle \)
In the second case, if D=D_{1}×⋯×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 pairwisecommuting, fullsized 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 F_{2} or \(F_{2} \times \mathbb{Z}\) or \(M_{\tau}=F_{2}\rtimes _{\tau}\mathbb{Z}\). It is obvious that every free subgroup of F_{2} 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 nontrivial 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 finiteindex 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 oneedge 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 oneedge splitting fixed by D, blowingup 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 blowup, 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 oneedge 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
where the action of Aut(F_{2}) 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(F_{2}) 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 F_{2} 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 fullsized 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 G_{1}×G_{2}×⋯×G_{k}. Suppose that rk_{F}(K)=l. Then, after reordering the factors, K is a normal subgroup of G_{1}×G_{2}×⋯×G_{l}×1×⋯×1.
Proof
Let g=(g_{1},g_{2},…,g_{k})∈K and without loss of generality assume g_{1}≠1. Let h∈G_{1} be an element that does not commute with g_{1}. Conjugation by (h,1,1,…,1) shows that (hg_{1}h^{−1},g_{2},g_{3},…,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 finiteindex subgroup of G then rk_{F}(H)=rk_{F}(G). If
is an exact sequence of groups then rk_{F}(G)≤rk_{F}(N)+rk_{F}(Q).
Recall that a group is fullsized if it contains a nonabelian free subgroup.
Proposition 2.3
Let G be a group generated by a set G_{1},…,G_{k} of commuting fullsized 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 fullsized subgroups in G/K.
Proof
As each G_{i} is fullsized, each G_{i} contains a group D_{i}≅F_{2}. Let
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=〈D_{1},…,D_{k}〉, which is isomorphic to D_{1}×D_{2}×⋯×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 D_{1}×D_{2}×⋯×D_{l}×1×⋯×1, and D_{i} embeds in G_{i}/K for i=l+1,…,k. Hence these quotient groups are also fullsized. □
2.2 The product rank of Aut(F _{N}) and Out(F _{N})
Fixing a basis \(\mathcal{B}= \{a_{1},a_{2}, x_{1},\ldots ,x_{N2}\}\) 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 {a_{1},a_{2}}, and let R_{i} be the subgroup consisting of automorphisms of the form [x_{i}↦x_{i}w, 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 \(D_{\mathcal{B}}= L_{1}\times R_{1}\times \cdots \times L_{N2}\times R_{N2}< {\mathrm{{Aut}}}(F_{N})\). 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
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 a_{1} and a_{2} 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
and by Lemma 2.2 we must have equality when N≥3. Since Aut(F_{2}) 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
For every N≥3 we have
Since \({\mathrm{{Out}}}(F_{2})\cong {\mathrm{{GL}}}(2,\mathbb{Z})\) is virtually free, rk_{F}(Out(F_{2}))=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 Gequivariant 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
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 ‘ϕtwistedly equivariant’. The map f_{ϕ} is unique. (This is true, more generally, for stabilizers of minimal irreducible Gtrees.) 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
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 Stab^{0}(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
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 torsionfree 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 Gorbit of \(T\in {\mathcal{FS}}\) is finite, then G fixes T; moreover G<Stab^{0}(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 oneedge splitting. A oneedge 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 fullsized 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 G_{1},…,G_{2N−4} of pairwisecommuting fullsized subgroups. Then G fixes a oneedge nonseparating free splitting of F_{N}.
Sketch proof
Let \(\mathcal{F}\) be a maximal, proper, Gperiodic 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 G_{1},…,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 G_{1},…,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 Ginvariant oneedge splitting T (there is a unique oneedge splitting where \(\mathcal{F}\) is elliptic). It is shown in the proof of [HW20, Theorem 6.1] that the stabilizer of a separating oneedge 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
so by uniqueness f_{ϕ}f_{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<Stab^{0}(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
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 (g_{1},…,g_{k};ϕ), the group operation is
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=F_{2}.
Our interest in M_{k}(F_{2}) 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, Stab^{0}(T)≅M_{2N−5}(F_{2}). 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}(F_{2}).
3.1 Generalities
We distinguish between the k visible copies of A in M_{k}(A) by writing M_{k}(A)=(A_{1}×⋯×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
where the last summand in A^{k+1} maps to
The existence of this embedding points to the fact A_{1}×⋯×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
Lemma 3.1
With the notation established above,

1.
(g_{1},…,g_{k},x)↦(g_{1}x^{−1},…,g_{k}x^{−1};ad_{x}) defines an isomorphism \(A^{k+1}\overset{\cong}{\to}M^{0}_{k}(A)\) with inverse (g_{1},…,g_{k};ad_{x})↦(g_{1}x,…,g_{k}x,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
defines an automorphism with the desired properties. □
3.2 Fixed subgroups, centralisers, and Nielsen transformations
We remind the reader that an automorphism τ∈Aut(F_{2}) is called a Nielsen transformation if there is a basis {x_{1},x_{2}} of F_{2} such that
Any two Nielsen transformations are conjugate in Aut(F_{2}) 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(F_{2}) consists entirely of powers of Nielsen transformations, then it is cyclic.
Proof
H intersects Inn(F_{2}) 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 nonzero 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 E_{1}. □
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(F_{2}) fixes a noncyclic subgroup of F_{2}, 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(F_{2}) is not cyclic, then its centraliser C(Λ)<Aut(F_{2}) is either trivial or else the cyclic subgroup generated by a Nielsen transformation.

2.
Let τ_{0},τ_{1}∈Aut(F_{2}) be Nielsen transformations. If 〈τ_{0}〉∩〈τ_{1}〉≠1, then \(\tau _{0}=\tau _{1}^{\pm 1}\).

3.
Let ϕ,ψ∈Aut(F_{2}), 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(F_{2}))=1.
Proof
This proof is a variation on the argument used by Gordon [Gor04, Theorem 3.2] to prove that rk_{F}(Aut(F_{2}))=1. If ϕ∈Aut(F_{2}) 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(τ_{0})=Fix(τ_{1}). Then apply (1) with Λ={ad_{w}:w∈Fix(τ_{0})}, 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(F_{2}) is virtually free, the centraliser of any nonabelian free subgroup is finite. So if there were a copy of F_{2}×F_{2} in Aut(F_{2}) then at least one of the factors, say F_{2}×1, would have to intersect Inn(F_{2}) nontrivially. The intersection would be normal in F_{2}×1, hence noncyclic. On the other hand, the centraliser of this intersection would contain 1×F_{2}, contradicting (1). □
Lemma 3.5
Let τ_{0},τ_{1}∈Aut(F_{2}) be Nielsen transformations. If some nonzero powers of τ_{0} and τ_{1} commute, then τ_{0} and τ_{1} commute, and \(\tau _{1}= {\mathrm{{ad}}}_{w}\tau _{0}^{\pm 1}\) for some w∈Fix(τ_{0}).
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 τ_{0} and τ_{1} commute, their eigenspaces coincide. Thus, replacing τ_{0} by its inverse if necessary, τ_{1}=ad_{w}τ_{0}. 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 τ_{0}(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(F_{2}) be a Nielsen transformation and suppose p≠0 and w∈F_{2}. Then ϕ=ad_{w}τ^{p} is a power of a Nielsen transformation if and only if w=yτ^{p}(y)^{−1} for some y∈F_{2}. Equivalently, ϕ=ad_{w}τ^{p} is a power of a Nielsen transformation if and only if there exists y∈F_{2} 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
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=F_{2}. To lighten the notation, we make the abbreviations M_{k}=M_{k}(F_{2}) and F^{k}:=(A_{1}×⋯×A_{k}). We have a fixed identification of each A_{i} with F_{2}, with respect to which the action defining the semidirect product M_{k}=F^{k}⋊Aut(F_{2}) is diagonal. Note that, for each index i, the subgroup \(A_{i}^{\tau }< A_{i}\) fixed by τ∈Aut(A_{i})=Aut(F_{2}) 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(F_{2}))=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}(F_{2}))≤k+1. The discussion in Sect. 3.1 gives an embedding of \(F_{2}^{k+1}\) in M_{k}(F_{2}), showing that the product rank is exactly k+1.
We want to classify embeddings D↪M_{k}, where
is generated by a collection of pairwise commuting subgroups D_{i} such that each D_{i} is fullsized. (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(F_{2}) 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 nontrivial 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
$$ D< A_{1}^{\tau}\times \cdots \times A_{j1}^{\tau}\times \langle A_{j},{ \underline{\tau }}\rangle \times A_{j+1}^{\tau}\dots \times A_{k}^{ \tau}\times J^{\tau}. $$
The following implicit feature of the theorem warrants explicit mention.
Addendum 3.9
If \({\overline{\pi }}(D_{i})\) is nontrivial 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 nontrivial, D might still conform to one of the descriptions in 3(iiii).

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 3manifold group in which free groups abound.
3.4 The proof of Theorem 3.8
Lemma 3.11
Consider D=〈D_{1},…,D_{k+1}〉↪M_{k}=F^{k}⋊Aut(F_{2}) with each D_{i} fullsized and [D_{i},D_{j}]=1 for all 1≤i<j≤k+1.

1.
There is a unique index i_{0} such that \(\pi _{D_{i_{0}}}\) is injective.

2.
There is a Nielsen transformation τ such that π(D_{j})<〈τ〉 for all j≠i_{0}, 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 i_{0}. As rk_{F}(Aut(F_{2}))=1, this index i_{0} 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 i_{0}=k+1. Then K=K_{1}×K_{2}×⋯×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 noncommuting elements \({\underline{u}}_{j},{\underline{v}}_{j}\in K_{j}\) for j=1,…,k and denote their nontrivial 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=(ω_{1},…,ω_{k};ϕ) be such. Since D_{k+1} commutes with K_{1}, we have \(g{\underline{u}}_{1} = {\underline{u}}_{1} g\), hence
whence \(\omega _{1} \phi (u_{1}) \omega _{1}^{1} = u_{1} \). Similarly, \(\omega _{1} \phi (v_{1}) \omega _{1}^{1} = v_{1} \). Since u_{1} and v_{1} 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 nonzero 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 y_{1}∈F_{2}.
Repeating this argument with {u_{j},v_{j}} in place of {u_{1},v_{1}}, 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
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(F_{2}) is fullsized and \({\overline{\pi }}(D_{k+1})<{\mathrm{{Out}}}(F_{2})\) is cyclic. Thus Λ:=π(D_{k+1})∩Inn(F_{2}) is not abelian, and Corollary 3.4 provides a Nielsen transformation τ that generates the centralizer of Λ in Aut(F_{2}). 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=τ_{0}.)
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 τ_{0}=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}=K_{0}⋊〈T_{0}〉 with K_{0}<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 K_{1}×K_{2}×⋯×K_{k}, so intersects A_{1}×A_{2}×⋯×A_{k} trivially. Hence C(D)<1⋊Aut(F_{2}). This subgroup of Aut(F_{2}) will fix the nonabelian subgroup K_{1}⊂A_{1}≅F_{2} 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 superficiallyexceptional 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 jcoordinate of \(T_{1} {\underline{w}}\) is \(t_{1j}\tau ^{p_{1}}(w)\), whereas the jcoordinate 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}∈F_{2} such that
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
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 \(m\in \mathbb{Z}\),
We now conjugate D by γ=(y_{21},y_{12},y_{13},…,y_{1k};1)^{−1}, noting that
where for j≥2 we have
Likewise,
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
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
The commutation [T_{1},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 [T_{2},K_{1}]=1 forces K_{1}<Fix(τ). Finally, the relations [T_{1},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 superficiallyexceptional 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 D_{1}<A_{1}⋊〈T_{1}〉 where T_{1}=(t_{1},1,…,1;τ^{p}) with p≠0. Again, the commutation [T_{1},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 pairwisecommuting 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(F_{2}). 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) fullsized 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) pairwisecommuting fullsized 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 nontrivial 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 e_{1},…,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_{i}b_{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 x_{1},…,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 x_{1},…,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
to the automorphism group of the rose is split surjective, and W_{k} is a set of coset representatives of Stab^{0}(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 Stab^{0}(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_{i}w(x_{i})v_{i}.
The group \(\widetilde{\mathrm{{Stab}}}(T)_{A}\) is isomorphic to
via the isomorphism
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_{j}b_{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_{j}v_{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 ϕtwistedly 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^{−1}h∈AXA. As ϕ(AXA)=AXA, the map f_{ϕ} preserves edges and so determines an isomorphism of T which is clearly ϕtwistedly equivariant. Conversely, if \(\phi \in \widetilde{\mathrm{{Stab}}}(T)_{A}\) then let f_{ϕ}:T→T be a ϕtwistedly 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
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_{j}A if and only if ϕ(x_{j})A=x_{j}A (equivalently, f_{ϕ} fixes this terminal vertex). As ϕ(x_{j})A=u_{j}x_{j}A, 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 Stab^{0}(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 F_{2}, so that \(\widetilde{\mathrm{{Stab}}}^{0}(T)_{A} \cong M_{2N4}(F_{2})\). 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 finiteindex 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≅F_{2} 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 x_{1} to x_{1}v_{1}, and sends each x_{i} to a word of the form u_{i}x_{i}v_{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 x_{1} with x_{1}v 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 (wx_{1})A(wx_{1})^{−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 u_{1}=1 in the above notation. Hence the intersection decomposes as an exact sequence
where k=N−2. The product rank of the kernel is then 2N−6 and the product rank of Aut(A)=Aut(F_{2}) 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 x_{1} (hence u_{1}=v_{1}=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 (wx_{1})A(wx_{1})^{−1}. Note that
as elements of G_{e} preserve \(x_{1}Ax_{1}^{1}\). If ϕ is also an element of G_{e′} then
This implies that ϕ(w)=w as w∈A. Therefore in the exact sequence
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 pairwisecommuting fullsized 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 D_{1},D_{2},…,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 〈D_{1},D_{2},…,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 fullsized 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 halfline towards this end, so that we can find a finitely generated subgroup \(\langle D_{1}',D_{2}',\ldots , D_{2N3}'\rangle \) with each \(D_{i}'\) a nonabelian free group fixing a halfline 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 e_{1},…,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 x_{1},…,x_{k−1}.
As above, take G=Stab^{0}(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∈Stab^{0}(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_{i}x_{i}b_{i} for some a_{i}∈A and b_{i}∈B.
It follows that Stab^{0}(T) fits in the exact sequence
Furthermore, this sequence splits so that Stab^{0}(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}(Stab^{0}(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 Gunrefinable if it admits no Ginvariant 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 Gunrefinable free splitting. If this splitting is nontrivial then it is a collapsed rose.

If T is the unique Gunrefinable 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 oneedge 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 Ginvariant 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 oneedge splittings to which T′ collapses). It follows that T is the unique Gunrefinable, Ginvariant 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 Ginvariant 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 Gunrefinable 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 Gunrefinable. Let \(\tilde{G}\) be the preimage of G in Aut(F_{N}). By Proposition 6.2 of [GH21], any two Gunrefinable 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 f_{0}:T→T_{0} followed by a morphism f_{1}:T_{0}→T_{1} that does not collapse edges. The map f_{1} is nontrivial as T and T′ are incompatible, which implies there exist edges e≠e′ based at the same vertex in T_{0} such that the edge paths f(e) and f(e′) have a common initial egde (if f_{1} were locally injective, then f_{1} would be an isomorphism). Partially folding e and e′ along a small initial segment gives a new \(\tilde{G}\) tree T_{1} with an extra \(\tilde{G}\)orbit of vertices at the fold. There is an induced morphism f_{2}:T_{1}→T′, which implies that any edge stabilizer of T_{1} fixes a nondegenerate arc in T′. Thus \(F_{N} < \tilde{G}\) has no nontrivial edge stabilizer in T_{1}, and T_{1} is a Ginvariant 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 \({{\mathrm{rk}_{F}}}({\mathrm{{Stab}}}_{{\mathrm{{Out}}}(F_{N})}(T)) \leq 2N5\).
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 oneedge separating splitting). As \(D < \mathrm{{\mathrm{IA}}_{N}(\mathbb{Z}}/\mathrm{3\mathbb{Z})}\), we have D<Stab^{0}(S). If A and B are free factor representatives of the vertex stabilizers in S then, as in Sect. 4.4, the group Stab^{0}(S) decomposes as a short exact sequence
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
and
Hence rk_{F}(Stab^{0}(S))≤(2N−2k−4)+2k−2=2N−6. If both A and B are cyclic (possibly trivial), then Stab^{0}(S) is virtually abelian, so rk_{F}(Stab^{0}(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
Similarly, if B is infinite cyclic then A is rank N−k and the same computation gives
□
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
is generated by 2N−4 pairwisecommuting fullsized subgroups D_{1},…,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 fullsized and let
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=〈D_{1},D_{2}〉 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 Dinvariant (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 oneedge 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
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 D_{3}/K,…,D_{2N−4}/K are fullsized in D_{A}. In particular, D_{A} is in the normalizer of the group generated by a commuting family of 2N−6 fullsized 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 x_{1}, 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(x_{1})=x_{1}a 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 D_{2},…,D_{2N−4} are fullsized in Im(f). Therefore Im(f) is contained in the normalizer of a commuting family of 2N−5=2(N−1)−3 fullsized 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 fullsized 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.

L_{1}×⋯×L_{N−2}×R_{1}×⋯×R_{N−2}×I(A)

\(L_{1}^{\tau }\times \cdots \times L_{N2}^{\tau }\times R_{1}^{\tau } \times \cdots \times R_{N2}^{\tau }\times I(A)^{\tau }\times \langle \tau \rangle \)

\(\langle L_{1}, \tau \rangle \times L_{2}^{\tau }\times \cdots \times L_{N2}^{\tau }\times R_{1}^{\tau }\times \cdots \times R_{N2}^{ \tau }\times I(A)^{\tau}\)

\(L_{1}^{\tau }\times \cdots \times L_{N2}^{\tau }\times R_{1}^{\tau } \times \cdots \times R_{N2}^{\tau }\times \langle I(A), \tau \rangle \)
In the second case, if D=D_{1}×⋯×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=〈a_{1},a_{2}〉 and stable letters x_{1},…,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
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 \(\widetilde{\mathrm{{Stab}}}^{0}(T)_{A}=M_{2N4}(A)\). 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.

L_{1}×⋯×L_{N−2}×R_{1}×⋯×R_{N−2}

\(L_{1}^{\tau }\times \cdots \times L_{N2}^{\tau }\times R_{1}^{\tau } \times \cdots \times R_{N2}^{\tau }\times \langle [\tau ] \rangle \)

\(\langle [\tau ], L_{1} \rangle \times L_{2}^{\tau }\times \cdots \times L_{N2}^{\tau }\times R_{1}^{\tau }\times \cdots \times R_{N2}^{ \tau}\)
In the second case, if D=D_{1}×⋯×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 Dinvariant 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 x_{1},…,x_{N−2}. In this case it is more natural to see Stab^{0}(T)≅M_{2N−5}(A) decomposing as the exact sequence:
so that Stab(T)=Stab^{0}(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 x_{1}A 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 x_{1}↦x_{1}v_{1} and for 2≤i≤N−2 maps x_{i}↦u_{i}x_{i}v_{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
Although the images of L_{2},…,L_{N−2} and R_{1},…,R_{N−2} are easy to see under Θ, the ‘natural’ representatives of the elements of L_{1} are not in \(\widetilde{\mathrm{{Stab}}}^{0}(T)_{e}\). However the transvection ϕ mapping x_{1}↦ax_{1} is equivalent in Out(F_{N}) to the automorphism ϕ′ sending x_{1} to x_{1}a and conjugating every other basis element by a^{−1}, so that \(\Theta (\phi ')=(a, \ldots , a; {\mathrm{{ad}}}_{a}^{1})\). It follows that L_{1} 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<Stab^{0}(T). The result then follows by combining Theorem 3.8 with the above isomorphism and the observation that Θ(L_{1})=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 finiteindex 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 finiteindex 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 onerelator 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 oneholed 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).
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 wellbehaved—it is hyperbolic and the fundamental group of a 3manifold 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 F_{2}=〈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(F_{2}) by identifying F_{2}⋊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 H_{1}<H_{2}<H_{3}<⋯ 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 F_{2} edge stabilizers are malnormal, so if a cylinder C_{g} contains more than one edge then g∉F_{2}. Hence g=ad_{x}τ^{k} for some x∈F_{2} and k≠0. The subgroup of F_{2} 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 blowup 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=〈a_{1},a_{2}〉, 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 F_{2} or a subgroup isomorphic to M_{τ}, and this enveloping group is a maximal direct product of fullsized subgroups of Aut(F_{N}) by Theorem D. Hence taking each F_{2} and a maximal free subgroup of the M_{τ} summand (which exists by Proposition 8.1) will suffice. □
References
Bestvina, M., Bridson, M.R.: Rigidity of the free factor complex (2023). arXiv:2306.05941
Bestvina, M., Guirardel, V., Horbez, C.: Boundary amenability of Out(F_{N}). Ann. Sci. Éc. Norm. Supér. (4) 55(5), 1379–1431 (2022)
Bestvina, M., Handel, M.: Train tracks and automorphisms of free groups. Ann. Math. (2) 135(1), 1–51 (1992)
Bass, H., Jiang, R.: Automorphism groups of tree actions and of graphs of groups. J. Pure Appl. Algebra 112(2), 109–155 (1996)
Birman, J.S., Lubotzky, A., McCarthy, J.: Abelian and solvable subgroups of the mapping class groups. Duke Math. J. 50(4), 1107–1120 (1983)
Brendle, T.E., Margalit, D.: Normal subgroups of mapping class groups and the metaconjecture of Ivanov. J. Am. Math. Soc. 32(4), 1009–1070 (2019)
Bridson, M.R., Pettet, A., Souto, J.: The abstract commensurator of the Johnson kernels. Preprint (2011)
Bridson, M.R., Wade, R.D.: Commensurations of Aut(F_{n}) and its Torelli subgroup. Geom. Funct. Anal. (2024)
Conant, J., Hatcher, A., Kassabov, M., Vogtmann, K.: Assembling homology classes in automorphism groups of free groups. Comment. Math. Helv. 91(4), 751–806 (2016)
Cohen, M.M., Lustig, M.: Very small group actions on \(\mathbbm{{}\mathbb{R}\mathbbm{}}\)trees and Dehn twist automorphisms. Topology 34(3), 575–617 (1995)
Collins, D.J., Turner, E.C.: All automorphisms of free groups with maximal rank fixed subgroups. Math. Proc. Camb. Philos. Soc. 119(4), 615–630 (1996)
Culler, M., Vogtmann, K.: Moduli of graphs and automorphisms of free groups. Invent. Math. 84(1), 91–119 (1986)
Farb, B., Handel, M.: Commensurations of Out(F_{n}). Publ. Math. IHES 105(1), 1–48 (2007)
Feighn, M., Handel, M.: Abelian subgroups of Out(F_{n}). Geom. Topol. 5(1), 39–106 (2011)
Guirardel, V., Horbez, C.: Boundaries of relative factor graphs and subgroup classification for automorphisms of free products (2019). arXiv:1901.05046
Guirardel, V., Horbez, C.: Measure equivalence rigidity of Out(F_{n}) (2021). arXiv:2103.03696
Guirardel, V., Levitt, G.: Deformation spaces of trees. Groups Geom. Dyn. 1(2), 135–181 (2007)
Guirardel, V., Levitt, G.: JSJ decompositions of groups. Astérisque, 395, vii+165 (2017)
Gordon, C.M.: Artin groups, 3manifolds and coherence. Bol. Soc. Mat. Mex. (3) 10(Special Issue), 193–198 (2004)
Handel, M., Mosher, L.: Subgroup decomposition in Out(F_{n}). Mem. Am. Math. Soc. 264(1280), vii+276 (2020)
Handel, M., Mosher, L.: Hyperbolic actions and 2nd bounded cohomology of subgroups of Out(F_{n}). Mem. Am. Math. Soc. 292(1454), v+170 (2023)
Hatcher, A., Vogtmann, K.: Homology stability for outer automorphism groups of free groups. Algebraic Geom. Topol. 4, 1253–1272 (2004)
Horbez, C., Wade, R.D.: Commensurations of subgroups of Out(F_{N}). Trans. Am. Math. Soc. 373(4), 2699–2742 (2020)
Ivanov, N.V.: Subgroups of Teichmüller Modular Groups. Translations of Mathematical Monographs, vol. 115. Am. Math. Soc., Providence (1992). Translated from the Russian by E. J. F. Primrose and revised by the author
Ivanov, N.V.: Automorphisms of complexes of curves and Teichmüller spaces. Int. Math. Res. Not. 14, 651–666 (1997)
Ivanov, N.V.: Mapping class groups. In: Handbook of Geometric Topology, pp. 523–633. NorthHolland, Amsterdam (2002)
Levitt, G.: Graphs of actions on Rtrees. Comment. Math. Helv. 69(1), 28–38 (1994)
Levitt, G.: Automorphisms of hyperbolic groups and graphs of groups. Geom. Dedic. 114(1), 49–70 (2005)
Serre, J.P.: Trees. Springer, Berlin (1980)
Scott, P., Swarup, G.A.: Regular neighbourhoods and canonical decompositions for groups. Astérisque, 289, vi+233 (2003)
Scott, P., Wall, T.: Topological methods in group theory. In: Homological Group Theory (Proc. Sympos., Durham, 1977), London Math. Soc. Lecture Note Ser., vol. 36, pp. 137–203. Cambridge University Press, Cambridge (1979)
Thurston, W.P.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Am. Math. Soc. (N.S.) 19(2), 417–431 (1988)
Acknowledgements
We are grateful to Mladen Bestvina, Sebastian Hensel and Camille Horbez for many stimulating conversations related to this work. We also offer our heartfelt thanks to the anonymous referee for an exceptionally careful reading of the manuscript and insightful comments that improved its exposition.
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The second author is supported by a University Research Fellowship from the Royal Society.
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Bridson, M.R., Wade, R.D. Direct Products of Free Groups in Aut(F_{N}). Geom. Funct. Anal. 34, 1337–1369 (2024). https://doi.org/10.1007/s00039024006885
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DOI: https://doi.org/10.1007/s00039024006885