Calculating the virtual cohomological dimension of the automorphism group of a RAAG

We describe an algorithm to find the virtual cohomological dimension of the automorphism group of a right-angled Artin group. The algorithm works in the relative setting; in particular it also applies to untwisted automorphism groups and basis-conjugating automorphism groups. The main new tool is the construction of free abelian subgroups of certain Fouxe-Rabinovitch groups of rank equal to their virtual cohomological dimension, generalizing a result of Meucci in the setting of free groups.


Introduction
Automorphism groups of right-angled Artin groups (or RAAGs) form a diverse and interesting family, encompassing the rich worlds of both integer matrix groups and automorphism groups of free groups. For any right-angled Artin group A Γ , Laurence [18] gave a generating set for Aut(A Γ ), and since this result authors have worked to understand higher finiteness properties of these groups. In particular, Charney and Vogtmann [7] showed that each outer automorphism group Out(A Γ ) has finite virtual cohomological dimension (vcd). Given recent constructions of classifying spaces for untwisted subgroups [6] and the natural analogs of congruence kernels for these groups [11], it is natural to ask what vcd(Out(A Γ )) actually is. Indeed, the construction of upper and lower bounds for specific examples and interesting subfamilies have been obtained in many cases [5,6,11,21], giving the vcd when these bounds coincide.
In this paper we give an algorithm to compute vcd(Out(A Γ )) for an arbitrary graph Γ. More generally, this algorithm gives the virtual cohomological dimension of any outer automorphism group of a right-angled Artin group relative to a collection of special subgroups. This includes the untwisted automorphism groups of [6] and partially symmetric (or basis-conjugating) outer automorphism groups of RAAGs.
In [11], the first and third authors initiated a study of relative outer automorphism groups of right-angled Artin groups, which are affectionately known as RORGs. Such a group is defined by taking collections G, H of special subgroups (a special subgroup is one of the form A ∆ given by an induced subgraph ∆ ⊂ Γ) of a right-angled Artin group A Γ and looking at the subgroup Out(A Γ ; G, H t ) of automorphisms that preserve each element of G and act trivially on each element of H (see Section 2.2). This approach is not an idle exercise in generalization if one wants to understand Out(A Γ ). The main result of [11] uses RORGs to construct a subnormal series for Out(A Γ ) (more generally, for an arbitrary RORG) such that the consecutive quotients of this series are either finite, free-abelian groups, copies of GL(n, Z), or groups known as Fouxe-Rabinovitch groups. We call such a normal series a decomposition series. We will see that the virtual cohomological dimension of Out(A Γ ) is the sum of the vcds of the consecutive quotients appearing in a decomposition series.
To make this process algorithmic, one needs to know how to find the vcd of a Fouxe-Rabinovitch group. Let us first recall the definition of these groups. Let G = G 1 * G 2 * · · · G k * F m be a free factor decomposition of a group G. An element Φ ∈ Out(G) belongs to the Fouxe-Rabinovitch group associated to this free factor decomposition if for each G i there exists a representative φ ∈ Φ restricting to the identity on G i . For example, the basis-conjugating automorphism group of a free group is the Fouxe-Rabinovitch group given by the free-factor decomposition F n = Z * Z · · · * Z. Going back to RAAGs, if each G i = A ∆i is a special subgroup, then the Fouxe-Rabinovitch group is the relative automorphism group Out(A Γ ; {A ∆i } t ).
Theorem A. Let A Γ = A ∆1 * A ∆2 * · · · * A ∆ k * F m be a free factor decomposition of a right-angled Artin group with k ≥ 1. Let d(∆ i ) be the size of a maximal clique in each ∆ i , and let z(∆ i ) be the rank of the center of A ∆i . Then There exists a free abelian subgroup of Out(A Γ ; {A ∆i } t ) of rank equal to the virtual cohomological dimension.
This generalizes a theorem of Meucci [20] on relative automorphism groups of free groups and Collins [9] on partially symmetric automorphism groups. To prove this theorem, we obtain a lower bound for the virtual cohomological dimension by constructing free-abelian subgroups of the appropriate rank. The upper bound is obtained by a careful analysis of simplex stabilizers for the action of the Fouxe-Rabinovitch group on the spine of relative Outer space (roughly speaking, we have to make sure that simplices of large dimension have small stabilizers). This uses work of Guirardel and Levitt [17].
In the case where k = 0, the virtual cohomological dimension of Out(F m ) was shown to be 2m − 3 in Culler and Vogtmann's seminal paper on Outer space [10]. There, the lower bound is obtained by finding a copy of Z 2m−3 in Out(F m ) generated by Nielsen automorphisms. The abelian subgroups found in the Fouxe-Rabinovitch case are very similar and made up of transvections and partial conjugations (see Remark 3.6). On the other side of the RAAG spectrum, similar results hold for GL(n, Z). Here the virtual cohomological dimension is equal to the Hirsch length of the polycyclic subgroup of upper triangular matrices. Given all of this, it is natural to conjecture that for an arbitrary RORG there is also a polycyclic subgroup of rank equal to the virtual cohomological dimension. Indeed, this conjecture holds in all known examples, but we cannot prove it in general. Luckily, we do not need explicit polycyclic subgroups to calculate vcd.
Theorem B. There is an algorithm which, given the input of a finite graph Γ and two collections of special subgroups G and H of A Γ , computes the virtual cohomological dimension of Out(A Γ ; G, H t ).
If one prefers to look at the absolute automorphism group, the vcds of Aut(A Γ ) and Out(A Γ ) differ by the dimension of A Γ /Z(A Γ ) (see Remark 4.4).
The main idea behind the proof of Theorem B is as follows. The existence of the polycyclic subgroups above imply that in all of these examples, the rational cohomological dimension of each group is equal to the vcd. Although vcd is only subadditive with respect to exact sequences, rational cohomological dimension is additive (by a theorem of Bieri [1]). Therefore the vcd of a RORG is the sum of the vcds of the pieces that appear in its decomposition series.
Structure of the paper. We describe the relevant background material on cohomological dimension and automorphism groups in Section 2. In Section 3 we give a proof of Theorem A and in Section 4 describe how the decomposition series of a RORG can be found algorithmically and complete the proof of Theorem B.

Cohomological dimension.
For a thorough treatment of cohomological dimension, the reader is referred to the books of Bieri [1] and Brown [3]. Let R be a unital commutative ring. For a group G, the cohomological dimension of G over R, denoted cd R (G) is given by The cohomological dimension of a group G is given by cd(G) = cd Z (G). The cohomological dimension satisfies cd R (G) ≤ cd(G) for any ring R. A group G is of finite type, or of type F , if G is the fundamental group of an aspherical CW-complex with a finite number of cells. If G is of finite type, then to find cd R (G) one only needs to look at the cohomology with coefficients in the group ring RG, and cd R (G) = max{n : H n (G; RG) = 0}.
If 1 → N → G → Q → 1 is an exact sequence of groups, then However, equality does not hold in general. For instance, Dranishnikov [13] constructed a family of hyperbolic groups G p such that cd(G p ) = 3 for all p, but cd(G p × G q ) = 5 whenever p = q. Roughly speaking, the failure of equality in (1) comes from torsion in the top cohomology group (this is explored in detail in [12]). Over a field these difficulties disappear, so that one has the following: is an exact sequence of groups of finite type, then Throughout this paper, we will be working with groups satisfying cd(G) = cd Q (G), and will be able to make use of the following proposition. If P is a torsion-free polycyclic group, then cd(P ) = cd Q (P ) = h(P ), where h(P ) is the Hirsch length of P -the number of infinite cyclic factors in a normal series for P (this follows from Proposition 2.2). If P is a subgroup of a group G then cd R (P ) ≤ cd R (G), so polycyclic groups can be used to find lower bounds for (rational) cohomological dimension. In particular, one has: In particular, if G has a finite-index torsion-free subgroup then vcd(G) = d.
Culler and Vogtmann [10] use the spine of Outer space and the existence of a free abelian subgroup of rank 2n − 3 to show that for any torsion-free, finite-index subgroup H of Out(F n ) one has Similarly, by combining Borel and Serre's calculation of the vcd [2] with the uppertriangular matrices in GL(n, Z), we see that for any torsion-free, finite index subgroup H of GL(n, Z).

RAAGs and RORGs.
Let A Γ be the right-angled Artin group determined by a finite graph Γ. If ∆ is a full subgraph of Γ we use A ∆ to denote the special subgroup generated by the vertices contained in ∆. An outer automorphism Φ of A Γ preserves A ∆ if there exists a representative φ ∈ Φ that restricts to an automorphism of A ∆ . An outer automorphism Φ acts trivially on A ∆ if there exists a representative φ ∈ Φ acting as the identity on A ∆ . If G, H are collections of special subgroups of A Γ , then the relative outer automorphism group Out(A Γ ; G, H t ) consists of automorphisms that preserve each A ∆ ∈ G and act trivially on each An important feature of RORGs is the fact that if A ∆ is invariant under the group Out(A Γ ; G, H t ) then there is a restriction homomorphism such that the image and kernel of R ∆ are also RORGs [11,Theorem E]. This allows us to study RORGs using inductive methods.
If G is a collection of special subgroups then a G-path in Γ is a sequence of vertices v 1 , . . . , v k of Γ such that each pair (v i , v i+1 ) either span an edge of Γ or are contained in some common element of G. A G-component of a subgraph ∆ ⊂ Γ is a maximal subgraph Λ ⊂ ∆ with the property that any two vertices in Λ are connected by a G-path in Λ. Given a vertex v ∈ Γ and a collection of special subgroups G we define G v to be the subset of G consisting of special subgroups that do not contain v, so that: There is a finite-index subgroup of the relative outer automorphism group, denoted Out 0 (A Γ ; G, H t ), that is generated by the inversions, transvections, and extended partial conjugations Out(A Γ ; G, H t ) contains (see Theorem D of [11]). These are defined below.
transvection of w on v is the automorphism ρ v w (respectively, λ v w ) that sends a vertex w to wv (respectively, vw) and fixes all other generators. This is a well-defined automorphism of • Extended partial conjugations. Fix a vertex v. If C is a union of components of Γ − st(v) then we define the extended partial conjugation π v C to send each w ∈ C to vwv −1 and fix all other generators. The au- The condition on extended partial conjugations is stated slightly differently to the one seen in [11,Proposition 3.9], where for developing the theory it was convenient to assume H (and often all special subgroups of groups in H) were contained in G. With the aim of simplifying computations, the above characterizations work in general.
We define the partial preorder . This is equivalent to the existence of an extended partial conjugation [π v C ] ∈ Out(A Γ ; G, H t ) which acts on A ∆ as a non-inner automorphism.
By using the above generating set of Out 0 (A Γ ; G, H t ), one can show the following (cf. [11,Lemma 2.2]).
is contained in some element of H and ∆ is not (G, H)-star-separated by any vertex of Γ.
Given the sets (G, H), we can use Proposition 2.4 to find the invariant special subgroups of Out 0 (A Γ ; G, H t ).

The virtual cohomological dimension of a Fouxe-Rabinovitch group
In this section we use the relative outer space of Guirardel and Levitt to find the virtual cohomological dimension of a Fouxe-Rabinovitch group associated to a free factor decomposition of a right-angled Artin group.
3.1. Fouxe-Rabinovitch groups and congruence subgroups. Let G = G 1 * G 2 * · · · * G k * F m be a free factor decomposition of a group. We let G = {G i } and define the Fouxe-Rabinovitch group associated to this free factor decomposition to be FR G = Out(G; G t ). This is the subgroup of Out(G) acting trivially on each G i in the decomposition. We do not assume the free factor decomposition is maximal: it need not be the Grushko decomposition of G. We do, however, require that this free factor decomposition is nontrivial in the sense that k ≥ 1 and k + m ≥ 2.
The level 3 congruence subgroup of FR G is defined in the same way as the subgroups of GL(n, Z) of the same name. It is the finite-index subgroup FR [3] G acting trivially on H 1 (G; Z/3Z). As the action of FR G on each G i is trivial, this is the same as the subgroup acting trivially on H 1 (F m ; Z/3Z) via the quotient map FR G → Out(F m ). If each G i and each G i /Z(G i ) is torsion-free, then so is each level 3 congruence subgroup of FR G (see [17,Thoerem 5.2]).

3.2.
Relative Outer space. Following the work of Guirardel and Levitt in [16,15], we recall the definition of the spine of the relative outer space given by a free factor decomposition of a group.
Definition 3.1. A Grushko G-tree is a minimal action of G on a simplicial tree T with trivial edge stabilizers such that each element of G is elliptic in T and each vertex stabilizer is either trivial or conjugate to an element of G. Two Grushko G-trees T 1 and T 2 are equivalent if there is a G-equivariant homeomorphism f : The set of Grushko G-trees forms a poset, where T 1 < T 2 if there is a (Gequivariant) subforest in T 2 which collapses to give the action of G on T 1 . The geometric realization of this poset is called the spine of relative Outer space and we will denote it by X G .
By a theorem of Guirardel and Levitt [16], the spine X G is contractible. The spine admits an action of Out(G; G) by precomposing the action of G on a tree T with the automorphism. Automorphisms may act nontrivially on each G i , but this action restricts to an action of FR G . Figure 1. A rose with m petals and k−1 leaves, whose Bass-Serre tree gives a vertex in the spine of relative Outer space.
Each n-simplex corresponds to a chain T 0 < T 1 < · · · < T n of Grushko G-trees. As the action of FR G preserves the number of edge orbits in a Grushko G-tree, the action of FR [3] G on X G is rigid (any automorphism preserving a simplex fixes it pointwise). The lemma below uses ideas from the proof of Proposition 3.7 of [17] and gives a description of simplex stabilizers for the action.
Lemma 3.2. Let σ be a simplex in X G corresponding to the chain T 0 < T 1 < · · · < T n of Grushko G-trees. Then the stabilizer of σ in FR where v i is the number of G i -orbits of edges at the vertex fixed by G i in T n and Proof. Firstly, as FR [3] G maps to a torsion-free subgroup of Out(F m ), automorphisms in FR [3] G preserving a Grushko G-tree T act trivially on the quotient graph T /G, and therefore preserve all collapses of T . This can be seen since the leaf vertices in T /G must have a non-trivial stabilizer in G, so must be in G. Hence the action fixes these vertices. Any such action induces a finite order element of Out(π 1 (T /G)) ∼ = Out(F m ), but since our image in here is torsion-free, the action must be trivial (note that if T /G is a circle, all vertices are fixed). As each T j is a collapse of T n , any automorphism in FR [3] G that fixes T n in FR [3] G also fixes each T j with j < n, so that: G , where Stab 0 FR G (T n ) denotes the stabilizer of T n in FR G that acts trivially on T n /G. This is the group of twists of the splitting [19, Section 2.4] and satisfies where, as in the hypothesis, each v i is the number of G i -orbits of edges in T n at the vertex fixed by G i .
It remains to justify the final inequality. Note that we have to collapse at least n orbits of edges in T n to obtain T 0 , so we may assume T n has N ≥ n orbits of vertices with trivial stabilizer. In total, the quotient graph T n /G has N + k vertices and N + k + m − 1 edges (as the fundamental group is F m ). There are at least 3 half-edges adjacent to each of the N vertices with trivial stabilizer. Therefore: as required.
As each v i ≥ 1, the inequality in Lemma 3.2 shows that n ≤ 2m + k − 2. It is not hard to check that the dimension of the spine is equal to 2m + k − 2 by exhibiting a graph of groups decomposition of G with 2m + k − 2 trivalent vertices with trivial stabilizers, trivial edge groups, and each nontrivial vertex group corresponding to a G i (see Figure 2). Figure 2. A graph of groups decomposition of G with a maximal number of edge orbits in the case that k = m = 4.
The above work allows one to bound the geometric dimension of a Fouxe-Rabinovitch group via the following theorem: Theorem 7.3.3). Let X be a contractible, rigid G-CW complex with dim(X) ≤ N . For each n, suppose that the stabilizer of each n-cell in X has geometric dimension at most d n . Then G has geometric dimension gd(G) ≤ max{d n + n : 0 ≤ n ≤ N }.
We will apply this to the specific case of right-angled Artin groups below.
3.3. Free product decompositions of RAAGs. For a finite graph Γ, we define d(Γ) to be the size of the largest clique in Γ. This is the same as the dimension of the Salvetti complex of A Γ . We define z(Γ) to be the number of vertices in Γ that are adjacent to every other vertex. This is the same as the rank of the center of A Γ , which is a finitely-generated free-abelian group. As FR [3] G is finite-index in FR G , the next theorem and its corollary imply Theorem A from the introduction. Theorem 3.4. Let FR G be the Fouxe-Rabinovitch group associated to a nontrivial free factor decomposition of a right-angled Artin group, and let FR [3] G be its level 3 congruence subgroup. Then gd(FR [3] Proof. As above, let X G be the spine of relative Outer space and let σ be a simplex of dimension n given by the chain of trees T 0 < T 1 · · · < T n . By Lemma 3.2, the stabilizer Stab FR [3] G (σ) of σ is a finite index subgroup of Therefore Theorem 3.3 implies that gd(FR [3] To establish equality it is enough to find a free abelian subgroup of FR G of rank equal to the right hand side of this equation. If we reorder the vertices so that A ∆1 has maximal dimension, we can find such a group inside the stabilizer of (the tree corresponding to) the rose given in Figure 1. In this case, Lemma 3.2 tells us that the stabilizer of the 0-cell given by the rose in X G under FR [3] G is a finite index subgroup of , which contains a free-abelian subgroup of the desired rank.
The following corollary is immediate from the proof: Corollary 3.5. There exists a free abelian subgroup of rank equal to gd(FR [3] G ), so that gd(FR [3] G ) = cd(FR [3] G ) = cd Q (FR [3] G ).
Remark 3.6. The free abelian subgroup used in the proof of Theorem 3.4 can be given quite explicitly. Firstly, order the factors of the decomposition so that A ∆1 has maximal dimension and let A i be the vertices of a maximal clique in ∆ i . Let X be the set of vertices generating the free factor F m . Then, take all left and right transvections ρ a x and λ a x for x ∈ X and a ∈ A 1 . Adding the partial conjugations of the subgroups A ∆j , for j > 1, by elements of A 1 gives a free abelian group of rank (2m + k − 1) · d(∆ 1 ) in Aut(A Γ ).
For each i = 1, . . . , k, and each vertex v ∈ A i , add the partial conjugation π v ∆i . Such partial conjugations are trivial if v is in the center of A ∆i , and since a maximal clique in ∆ i must contain all vertices in the center, this gives us d(∆ i )−z(∆ i ) partial conjugations for each i. One can check that all of automorphisms above generate a free abelian subgroup of Aut(A Γ ) of rank The only inner automorphisms that appear in the above group come from products of generators with acting letter a ∈ A 1 , so that the intersection of this subgroup with the inner automorphisms has rank d(∆ 1 ). Subtracting this gives the rank in Out(A Γ ).

Calculating the vcd
We now give the details of the algorithm to compute the vcd of a RORG. As a first step, we explain how the decomposition procedure for a RORG given in [11] is algorithmic.
(D1) a finitely generated free abelian group, (D2) isomorphic to GL(m, Z), or (D3) a Fouxe-Rabinovitch group given by a free factor decomposition of a special subgroup of A Γ .
Note that Out(F m ) may arise as a quotient via case (D3). As in the introduction, we call such a subnormal series a decomposition series for the group.
The most natural way to find the consecutive quotients in a decomposition series is to first build a decomposition tree for Out 0 (A Γ ; G, H t ). This is a rooted tree where every internal vertex is labelled by a group of the form Out 0 (Γ v ; G v , H v ), with Γ v a subgraph of Γ and G v , H v sets of special subgroups of A Γv . Our initial group is at the root. Each internal vertex G v has two descendants K v and I v forming an exact sequence Every leaf of this tree is labelled by a group of the form (D1), (D2), or (D3) and one can show (e.g. using induction on the size of the tree) that the leaves of the tree give consecutive quotients in a subnormal series for the root. An example of such a tree is given in [11, Figure 6].

Proposition 4.2.
There is an algorithm that produces a decomposition tree for Out 0 (A Γ ; G, H t ).
Proof. The process for obtaining a tree is iterative. Given a vertex v in the tree, labelled by Out 0 (Γ v ; G v , H t v ), we describe below how to either (1) recognise Out 0 (Γ v ; G v , H t v ) as a group of type (D1), (D2), or (D3), or (2) extend the tree by adding two edges and two new descendants v 1 and v 2 of v, such that the group labelling v 1 and v 2 is either a RORG of lower complexity (see [11,Theorem 5.9] for details on the complexity), or a group of type (D1), (D2), or (D3).
If a new vertex has not been recognised as (D1), (D2), or (D3), then we repeat the process on this vertex. Because the complexity of RORGs decreases as we get further from the root, this algorithm will terminate.
and (H v ) ∆ is defined similarly. This image is nontrivial if and only if there is an inversion, extended partial conjugation, or transvection with nontrivial image under R ∆ . This is a finite list of elements, and checking if each one has nontrivial image is a simple process. We now divide into cases according to the nature of the images of restriction maps. Case 1. There is a restriction map R ∆ with nontrivial image.
In this case we use the exact sequence given by [11,Theorem E]. As per the proof of [11,Theorem 5.9], the complexity of the RORG in the kernel and quotient is strictly lower than that of Out 0 (Γ v ; G v , H t v ). In this case we add two new descendants below v in the tree, with vertices labelled by the kernel and image above. Case 2. All restriction maps have trivial image.
As in [11,Section 5], we can break into five subcases. Here is a Fouxe-Rabinovitch group where F m is a free group on m isolated vertices not contained in any element of G v and the A ∆i are the remaining G v -connected components ([11, Proposition 5.2]).
The vertices which (G v , H v )-star-separate form a complete graph Θ and, as per the proof of [11,Proposition 5 is a free-abelian group of rank equal to |Θ|. Case 2c. Γ v is connected and the center Z(A Γv ) of A Γv is trivial.
In this case Out 0 (Γ v ; G v , H t v ) is generated by commuting partial conjugations with acting letters v that have N v (G, H)-connected components. It is not hard to check (e.g. using the first Johnson homomorphism [23]) that these elements form a free-abelian group of rank (N v − 1).
Case 2d. Γ v is connected and Z(A Γv ) is a proper, nontrivial subgroup.
If ∆ = Γ v −Z(Γ v ) we apply [11,Proposition 5.6]. There is a projection homomorphism P ∆ with image Out 0 (A ∆ ; (G v ) t ∆ ) (with (G v ) ∆ as defined above) whose kernel is a free abelian group with basis given by the leaf transvections in Out 0 (A Γ ; G, H t ). These are transvections ρ u w with w ∈ Z(Γ) and u / ∈ Z(Γ), see [8]. It is a quick check on each transvection to determine if it is a leaf transvection. We therefore add two descendants below v, one labelled by a free abelian group of the appropriate rank, and the other labelled by Out 0 (A ∆ ; (G v ) t ∆ ). Case 2e. Γ v is complete and A Γv = Z n for some n.
It is described in [11,Proposition 5.8] how the group fits in the exact sequence where A is a finitely generated free abelian group of matrices, so that the rank is easy to compute. We thus add two descendants below v, one labelled by A and the other by GL(m, Z).
Note that the construction of a decomposition tree involves many choices, as at each step we only pick some invariant special subgroup A ∆ for which there is a restriction map. In this direction, Brück [4, Section 7] uses careful choices of restriction maps to construct a decomposition tree for Out 0 (A Γ ; G, H t ) where the leaves can be described quite explicitly. As a trade-off, the leaves that appear in the decomposition tree of Brück are slightly more general (there are groups generated by partial conjugations that are not necessarily of type (D1), (D2), or (D3)).

4.2.
Completing the proof of Theorem B. To complete the proof of Theorem B, we describe how to compute the vcd of a RORG step-by-step: Step 1: Build a decomposition tree for Out 0 (A Γ ; G, H t ). This is detailed in Proposition 4.2.
Step 2: Find the vcd of each leaf. Each leaf is free-abelian, a copy of GL(n, Z), or a Fouxe-Rabinovitch group, so this can be read off via the calculations of Borel-Serre [2] and Culler-Vogtmann [10] discussed in Section 2.1 and Theorem A.
Step 3: Add the vcds of the leaves to find the vcd of the root. We do not need to explain how to carry out this step, but we should justify why it works. This is where the discussion of rational cohomological dimension given in Section 2.1 comes into play. The key point here is that we can restrict to the congruence subgroup Out [3] (A Γ ; G, H t ) of Out 0 (A Γ ; G, H t ). This is the torsion-free, finite-index subgroup given by the elements acting trivially on H 1 (A Γ , Z/3Z). By [11,Theorem 4.8], the short exact sequence coming from each projection map restricts to a short exact sequence 1 → Out [3] R∆ −→ Out [3] (A ∆ ; (G v ) ∆ , (H v ) t ∆ ) → 1. for congruence subgroups. Similar behaviour happens with the projection maps that appear in Case 2d and Case 2e during the construction of the decomposition tree (one can see this as both of the projection maps split). As a result, one obtains an analogous decomposition tree for Out [3] (A Γ ; G, H t ) where each vertex is a level three congruence subgroup of the corresponding vertex in the decomposition tree for Out 0 (A Γ ; G, H t ). This gives a subnormal series 1 = H 0 < H 1 < H 2 < · · · < H K = Out [3] (A Γ ; G, H t ) of Out [3] (A Γ ; G, H t ) where the consecutive quotients are congruence subgroups of the leaves of the decomposition tree for Out 0 (A Γ ; G, H t ) (and the leaves given by free-abelian groups are still free-abelian of the same rank). Some leaves, in particular those isomorphic to GL(1, Z) = Z/2Z, will now be trivial. All of these groups are of finite type, and have rational cohomological dimension equal to their cohomological dimension (using either the discussion in Section 2.1 or Theorem A). By Bieri's theorem (Theorem 2.1) and Proposition 2.2, the sum of the (rational) cohomological dimensions of the leaves is equal to the cohomological dimension of Out [3] (A Γ ; G, H t ), justifying the calculation of the vcd of Out 0 (A Γ ; G, H t ) given above.
Remark 4.4. The above work shows that the rational cohomological dimension of a (relative) outer automorphism group is the same as its cohomological dimension. As the inner automorphisms are isomorphic to A Γ /Z(A Γ ), the same is true for Inn(A Γ ). Bieri's theorem implies that the vcds of Out(A Γ ) and Aut(A Γ ) differ by the dimension of A Γ /Z(A Γ ).