Between buildings and free factor complexes: A Cohen–Macaulay complex for Out(RAAGs)

For every finite graph Γ$\Gamma$ , we define a simplicial complex associated to the outer automorphism group of the right‐angled Artin group (RAAG) AΓ$A_\Gamma$ . These complexes are defined as coset complexes of parabolic subgroups of Out0(AΓ)$\operatorname{Out}^0(A_\Gamma )$ and interpolate between Tits buildings and free factor complexes. We show that each of these complexes is homotopy Cohen–Macaulay and in particular homotopy equivalent to a wedge of d$d$ ‐spheres. The dimension d$d$ can be read off from the defining graph Γ$\Gamma$ and is determined by the rank of a certain Coxeter subgroup of Out0(AΓ)$\operatorname{Out}^0(A_\Gamma )$ . To show this, we refine the decomposition sequence for Out0(AΓ)$\operatorname{Out}^0(A_\Gamma )$ established by Day–Wade, generalise a result of Brown concerning the behaviour of coset posets under short exact sequences and determine the homotopy type of free factor complexes associated to relative automorphism groups of free products.


INTRODUCTION
In addition to Theorem A, we show that  has the following properties which indicate that it is a reasonable analogue of Tits buildings and free factor complexes.

Properties of 
• Building. If = GL (ℤ), the complex  is isomorphic to the building associated to GL (ℚ) (Proposition 4.3). • Free factor complex. If = Out( ), the complex  is isomorphic to the free factor complex associated to (Proposition 4.6). • Cohen-Macaulayness.  is homotopy Cohen-Macaulay and in particular a chamber complex (Proposition 8.2). • Facet-transitivity. Any maximal simplex of  forms a fundamental domain for the action of (Section 3.1.3). • Stabilisers. The vertex stabilisers of this action are exactly the conjugates of the elements of ( ). Stabilisers of higher dimensional simplices are given by the intersections of such conjugates and can be seen as parabolic subgroups of lower rank (Section 8.2). • Parabolics as relative automorphism groups. Every maximal standard parabolic ∈ ( ) is itself a relative automorphism group of the form Out 0 ( Γ ; ,  ) and rk( ) = |( )| = rk( ) − 1 (Proposition 8.4). • Rank via Weyl group. Similar to a group with BN-pair, the rank rk( ) is equal to the rank of a naturally defined Coxeter subgroup Aut 0 (Γ) ⩽ (Corollary 8.9). • Direct and free products. The construction is well-behaved under taking direct and free products of the underlying RAAGs, that is, under passing from Out 0 ( Γ ) to Out 0 ( Γ × Γ ′ ) or to Out 0 ( Γ * Γ ′ ) (Section 7.2.3).
Remark 1.1. This article focuses on topological properties of . This is the perspective that makes the complexes it generalises look quite similar, reflecting the fact that Out( ), GL (ℤ) and mapping class groups of surfaces share many homological properties (see, for example, [17] and references therein). It should be noted that from a geometric perspective, there are significant differences between the associated complexes: While the free factor complex [3] and the curve complex [34] are hyperbolic, spherical buildings such as the one associated to GL (ℤ) have finite diameter. In general, Out( Γ ) has many aspects of GL (ℤ), so one should probably not expect associated complexes to have a 'purely hyperbolic' flavour (cf. the work of Haettel [25]).
get a better control on the induction steps that are needed and to get a more explicit description of the resulting base cases. An overview of this can be found in Section 7.1. To make use of this inductive procedure, we establish a theorem regarding the behaviour of coset posets and complexes under short exact sequences. This generalises results of Brown [8], Holz [31] and Welsch [46] and can be phrased as follows.
Theorem C. Let be a group,  a family of subgroups of and ⊲ a normal subgroup. If  is strongly divided by , there is a homotopy equivalence CC( , ) ≃ CC( ∕ , ) * CC( ,  ∩ ).
Here, * denotes the join on geometric realisations,  and  ∩ are certain families of subgroups of ∕ and , respectively, and being strongly divided by is a compatibility condition on the family . (For the definitions, see Section 3.2; for an explicitly stated special case of Theorem C that we will use in this article, see Corollary 3.19.) Note that if two spaces and are homotopy equivalent to wedges of spheres, then so is their join * . Thus, combining Theorem C with the decomposition of Day-Wade, we are able to reduce the question of sphericity of  to the cases where is either isomorphic to GL (ℤ) or a Fouxe-Rabinovitch group. In the former case, sphericity follows from the Solomon-Tits theorem (Proposition 4.3). In the latter case, we are led to study relative versions of free factor complexes (see Definition 4.4) and show the following.
Theorem D. Let = * 1 * ⋯ * be a finitely generated group. Then the complex of free factors of relative to { 1 , … , } is homotopy equivalent to a wedge of spheres of dimension − 2.
In the case where the group is a RAAG, Theorem D is a special case of Theorem A. However, we prove it without making this assumption by using the techniques of [10].

Structure of the article
Many sections of this article can be read independently from the others. We start in Section 2 by recalling some well-known results from topology that we will use throughout the paper. Section 3 contains definitions and basic properties of coset complexes and higher generating subgroups as well as the proof of Theorem C; it can be read completely independently from the rest of this text. The reader not so much interested in details about coset complexes might just want to skim Section 3.1 and then have a look at Corollary 3.19, which summarises the results of this section in the way they will be used later on. In Section 4, we give a definition of the building associated to GL (ℤ) and determine the homotopy type of relative free factor complexes (Theorem D). A reader willing to take this on faith may just have a look at the main results of this section, namely, Theorems 4. 20 and 4.21; the general theory of automorphisms of RAAGs is still not needed for this. Section 5 contains background about (relative) automorphism groups of RAAGs. Section 6 is in some sense the core of this article: We define (maximal) parabolic subgroups and the rank of and combine the results of the previous sections in order to prove Theorem A. In Section 7, we summarise to which extent we can refine the inductive procedure of Day-Wade and give examples of our construction for specific graphs Γ. In Section 8.1, we show Cohen-Macaulayness of , define parabolic subgroups of lower rank and prove Theorem B. We then show how the dimension of our complex is related to the rank of a Coxeter subgroup of (see Corollary 8.9). We close with comments about the limitations of our construction and open questions in Section 9.

Posets and their realisations
Let = ( , ⩽) be a poset (partially ordered set). If ∈ , the sets ⩽ and ⩾ are defined by Similarly, one defines < and > . For , ∈ , the open interval between and is defined as A chain of length in is a totally ordered subset 0 < 1 < ⋯ < . For each poset = ( , ⩽), one has an associated simplicial complex Δ( ) called the order complex of . Its vertices are the elements of and higher dimensional simplices are given by the chains of . When we speak about the realisation of the poset , we mean the geometric realisations of its order complex and denote this space by ‖ ‖ ∶= ‖Δ( )‖. By an abuse of notation, we will attribute topological properties (for example, homotopy groups and connectivity properties) to a poset when we mean that its realisation has these properties.
The join of two posets and , denoted * , is the poset whose elements are given by the disjoint union of and equipped with the ordering extending the orders on and and such that < for all ∈ , ∈ . The geometric realisation of the join of and is homeomorphic to the topological join of their geometric realisations: The direct product × of two posets and is the poset whose underlying set is the Cartesian product {( , ) | ∈ , ∈ } and whose order relation is given by ( , ) ⩽ × ( ′ , ′ ) if ⩽ ′ and ⩽ ′ .
A map ∶ → between two posets is called a poset map if ⩽ implies ( ) ⩽ ( ). Such a poset map induces a simplicial map from Δ( ) to Δ( ) and hence a continuous map on the realisations of the posets. It will be denoted by ‖ ‖ or just by if what is meant is clear from the context.

Fibre theorems
An important tool to study the topology of posets is given by so-called fibre lemmas comparing the connectivity properties of posets and by analysing the fibres of a poset map between them. The first such fibre theorem appeared in [36, Theorem A] and is known as Quillen's fibre lemma.
The following result shows that if one is given a poset map such that the fibres have only vanishing homotopy groups up to a certain degree, one can also transfer connectivity results between the domain and the image of . Recall that for ∈ ℕ, a space is -connected if ( ) = {1} for all ⩽ and is (−1)-connected if it is non-empty. [37,Proposition 7.6]. Let ∶ → be a poset map such that the fibre −1 ( ⩽ ) is -connected for all ∈ . Then is -connected if and only if is -connected.

Lemma 2.2
For a poset = ( , ⩽), let = ( , ⩽ ) be the poset that is defined by ⩽ ∶ ⇔ ⩽ . Using the fact that one has a natural identification Δ( ) ≅ Δ( ), one can draw the same conclusion as in the previous lemmas if one shows that −1 ( ⩾ ) is contractible or -connected, respectively, for all ∈ .
Another standard tool which is helpful for studying the topology of posets is as follows.
Later on, we will mostly use the following consequence of this lemma.
Proof. Without loss of generality, assume that ( ) ⩽ for all ∈ . Let ∶ ′ ↪ denote the inclusion map. Then for all ∈ , we have • ( ) ⩽ , so by Lemma 2.3, this composition is homotopic to the identity. As • = id ′ , the inclusion is a homotopy equivalence and the claim follows from [29,Proposition 0.19]. □

Spherical complexes and their joins
Recall that a topological space is -spherical if it is homotopy equivalent to a wedge of -spheres; as a convention, we consider a contractible space to be homotopy equivalent to a (trivial) wedge of -spheres for all and the empty set to be (−1)-spherical. By the Whitehead theorem, an -dimensional CW-complex is -spherical if and only if it is ( − 1)-connected. Furthermore, sphericity is preserved under taking joins.
The word 'homotopy' here refers to the original 'homological' notion of being 'Cohen-Macaulay over a field '. This homological condition is weaker than the homotopical one and came up in the study of finite simplicial complexes via their Stanley-Reisner rings (see [43]). For more details on Cohen-Macaulayness and its connections to other combinatorial properties of simplicial complexes, see [4].

Standing assumptions
Throughout this section, let be a group and let  be a family of proper subgroups of . (1) The coset poset CP( , ) ∶= ( , ⊆) is the partially ordered set consisting of the elements of  , where g 1 1 ⩽ g 2 2 if and only if g 1 1 ⊆ g 2 2 .

Background and relation between poset and complex
(2) The coset complex CC( , ) is the nerve ( ) of the covering of given by  .
In this form, coset complexes were introduced by Abels and Holz in [1] but they appear with different names in several branches of group theory: The main motivation of Abels and Holz was to study finiteness properties of groups. Recent work in this direction can be found in the work of Bux, Fluch, Marschler, Witzel and Zaremsky [11] and Santos-Rego [39]. In [35], Meier, Meinert and VanWyk used these complexes to study the BNS invariants of right-angled Artin groups. Well-known examples of coset posets are given by Coxeter and Deligne complexes [14]. Brown [8] studied the coset poset of all subgroups of a finite group and its connection to zeta functions. Generalisations of his work can be found in the articles of Ramras [38] and Shareshian and Woodroofe [41]. However, the examples that are most important to the present work are given by Tits buildings and free factor complexes (see Section 4).
The order complex of the coset poset CP( , ) has the same vertices as the coset complex CC( , ) but the higher dimensional simplices do not have to agree (see Figure 1). However, if we assume that  be closed under finite intersections, the topology of these complexes is the same. Lemma 3.3. Suppose that 1 , 2 ∈  implies 1 ∩ 2 ∈ . Then CC( , ) deformation retracts to CP( , ). In particular, we have The left-hand side shows CC(ℤ, ) and CP(ℤ, ), the right-hand side CC(ℤ,) and CP(ℤ,), where  = {2ℤ, 3ℤ} and = {2ℤ, 3ℤ, 6ℤ}. In both pictures, the coset poset is drawn in black and the coset complex is obtained from it by adding the blue parts Proof. As  is closed under intersections, the intersection of two cosets from  is either empty or also an element of  . Hence, we can define a map from the poset of simplices of CC( , ) to CP( , ) by sending (g 0 0 , … , g ) to ⋂ g . On the corresponding order complexes, this defines a deformation retraction from the barycentric subdivision of CC( , ) to Δ CP( , ) (see [1,Theorem 1.4(b)]). See Figure 1 for an easy example. □ Let denote the family consisting of all finite intersections of elements from . The following was proved by Holz in [31].
Proof. The nerve ( ) of a collection  of subsets of is homotopy equivalent to the simplicial complex whose simplices are the non-empty finite subsets of contained in some ∈  (see [1, Theorem 1.4(a)]). This implies the first claim. The second statement is an immediate consequence of the first one. □ Remark 3.5. The preceding lemmas imply that for any family  of subgroups of , we have It follows that we can always replace a coset complex by a coset poset. The advantage of this is that it allows us to apply the tools of poset topology, for example, the Quillen fibre lemma, to study the topology of these complexes. The trade-off, however, is that we have to increase the size of our family of subgroups.

Higher generation
We now turn our attention to coset complexes. Definition 3.6. The free product of  amalgamated along its intersections is the group given by Definition 3.7. We say that  is -generating for if CC( , ) is ( − 1)-connected, that is, (CC( , )) = {1} for all < .
The term 'higher generating subgroups' was coined by Holz in [31] and is motivated by the following theorem.
(2)  is 2-generating if and only if is the free product of  amalgamated along its intersections.
Roughly speaking, the latter means that the union of the subgroups in  generates and that all relations that hold in follow from relations in these subgroups. The concept of 3-generation has a similar interpretation using identities among relations (see [1, 2.8]).

Group actions and detecting coset complexes
Coset complexes are endowed with a natural action of given by left multiplication. These complexes are highly symmetric in the sense that this action is facet transitive: Assume that  is finite. Then CC( , ) has dimension || − 1 and  itself is the vertex set of a facet, that is, a maximal simplex, of the coset complex. This (and hence any other) facet is a strict fundamental domain for the action of . The following converse of this observation is due to Zaremsky. Proposition 3.9 (see [11,Proposition A.5]). Let be a group acting by simplicial automorphisms on a simplicial complex , with a single facet as a strict fundamental domain. Let Then the map is an isomorphism of simplicial -complexes.

Short exact sequences
We will later on study coset complexes in the setting where = Out( Γ ), the outer automorphism group of a right-angled Artin group. For this, we want to use the decomposition sequences of Out( Γ ) developed in [19]. To do so, we need to study the following question: If fits into a short exact sequence, can the coset complex CC( , ) be decomposed into 'simpler' complexes related to the image and kernel of the sequence? There is a special case where this question can easily be answered.

Coset complexes and direct products
Assume that we have a group factoring as a direct product = 1 × 2 and let  be a family of subgroups such that each ∈  contains either {1} × 2 or 1 × {1}; denote the set of those elements of  satisfying the former by  1 and the set of those satisfying the latter by  2 . Now given 1 , ′ 1 ∈  1 , we have where 1 is the projection map → 1 . The analogous statement holds for 2 , ′ 2 ∈  2 . On the other hand, if we take 1 ∈  1 and 2 ∈  2 , all of their cosets intersect non-trivially because It follows that the coset complex CC( , ) decomposes as a join CC( , ) ≅ CC( 1 , 1 ( 1 )) * CC( 2 , 2 ( 2 )).
However, the situation becomes more complicated if we consider semi-direct products or general short exact sequences References [31,Proposition 5.17;46,Theorem 7.3;8,Proposition 10] contain results in this direction for the cases where every ∈  is a complement of , every ∈  contains and where is a finite group and  is the set of all subgroups of , respectively. Our work in this section provides a common generalisation of all three of these results (see Theorem 3.18).

Notation and standing assumptions
From now on, we will fix a normal subgroup ⊲ and assume that  is a set of proper subgroups of . In this situation, we can write  as a disjoint union  =  ⊔  , where For elements g ∈ and subgroups ⩽ of , letḡ and denote the image of g and in the quotient ∕ , respectively. The family  gives rise to a family of proper subgroups of ∕ , denoted by Similarly,  gives rise to a family of proper subgroups of , denoted by

Coset posets and short exact sequences
We start by considering the behaviour of coset posets under short exact sequences.
Definition 3.10. The family  of proper subgroups of is divided by if the following holds true.
In what follows, we will use the following elementary observations. Lemma 3.11. Let , ⩽ be two subgroups of and assume that = . Then one has ( ∩ ) ⋅ = .
Proof. As = , we can write g = with ∈ and ∈ . The intersection The next proposition is a generalisation of [8,Proposition 10]. Our proof closely follows the ideas of Brown. such that restricts to the identity on and (g ) =ḡ for all g ∈ . As no coset from can be contained in a coset from , this map is order-preserving, that is, a poset map. For ∈ CP( ∕ , ) * , define to be the fibre of with respect to . We want to use Lemma 2.1 to show that is a homotopy equivalence. For this, we need to show that is contractible. If ∈ CP( ∕ , ), this is clear: Write =ḡ such that g ∈ , ∈  . As divides , the subgroup is contained in  and g ⋅ is the unique maximal element of . This immediately implies contractibility of . Now assume ∈ . Using the natural action of on these posets, we can assume that = ∈  . By definition of the join, the poset can as be written as = ∪ ⩽ . On the level of geometric realisations, it decomposes as where ′ ∶= ∩ ⩽ is equal to ( ) ⩽ . (To see this, note that no coset from can be contained in a coset from and that if g ∈ is contained in some g ′ ′ ∈ ⩽ , we have g ∈ ′ .) Next, we show that ‖ ′ ‖ is a strong deformation retract of ‖ ‖. This implies that is homotopy equivalent to ⩽ , which is contractible as it has as unique maximal element.
The poset ′ is given by all cosets g ⊆ such that ∈  . Hence, Lemma 3.12 implies that for g ∈ , the intersection (g ⋅ ) ∩ is an element of ′ . This allows us to define poset maps For g ∈ ′ , we have g ⊆ , hence If on the other hand g ∈ , one has by Lemma 3.12 for some ∈ g ⋅ ∩ . By Lemma 3.11, we have ( ∩ ) ⋅ = , so it follows that • (g ) = ⋅ ⊇ g . Lemma 2.3 now implies that and are homotopy equivalences which are inverse to each other. Furthermore, we have g ⊆ (g ) for all g ∈ ′ , so again by Lemma 2.3, the map is homotopic to the inclusion ′ ↪ which must hence be a homotopy equivalence as well. It follows that ‖ ′ ‖ is a strong deformation retract of ‖ ‖. □

Coset complexes and short exact sequences
We will now translate the results obtained in the last section to coset complexes. The following observation follows from elementary group theory.
We obtain the following relation between CC( ,  ) and CC( ,  ∩ ). Proof. As = = for all ∈  , each vertex of CC( ,  ) can be written as with ∈ . Use this to define the map which we claim is an isomorphism of simplicial complexes.
As ∈ , this map is well-defined on vertices. It also clearly is surjective on vertices. Now assume that for 1 , 2 ∈ and 1 , 2 ∈  , one has 1 ⋅ 1 ∩ = 2 ⋅ 2 ∩ . As the two cosets coincide, so do the subgroups 1 ∩ = 2 ∩ . By Lemma 3.14, this implies that 1 = 2 . It follows in particular that 1 1 = 2 2 which shows that defines a bijection between the vertex sets of the two coset complexes.
Using Lemma 3.11, it is easy to see that every family of subgroups which is strongly divided by is also divided by . On top of that, given a family which is strongly divided, we can even produce a family which is closed under intersections and still divided by as the following lemma shows. Recall that denotes the family of all finite intersections of elements from . Lemma 3.17. If  is strongly divided by , the family is divided by . Furthermore, we have the following.
1. is equal to the family of all finite intersections of elements from  , that is, 2. The image of in ∕ is equal to the family of finite intersections of elements from , that is, Proof. Every˜∈ can be written as where for all and , one has ⊆ and ∈  . If˜∈ , we must have = 0, that is,˜= 1 ∩ ⋯ ∩ is a finite intersection of elements from  . On the other hand, every such finite intersection forms an element of because one has ( 1 ∩ ⋯ ∩ ) = , which proves Item 1. This also implies that if˜∈ , we have ⩾ 1. It follows from Lemma 3.11 that˜is equal to 1 ∩ ⋯ ∩ . This is a finite intersection of elements from  and hence contained in. Furthermore, this implies that the image of˜in ∕ is equal to = 1 ∩ ⋯ ∩ , showing Item 2.
The last thing that remains to be checked is that˜is divided by , that is, that for all˜∈  and˜∈ , one has˜∩˜∈. However, we already know that˜= 1 ∩ ⋯ ∩ , sõ ∩˜is itself a finite intersection of elements from . □ We are now ready to prove Theorem C which we restate as follows. On the other hand, Lemma 3.17 also tells us that consists of all finite intersections of elements from  . It follows that CP( , ) ≃ CC( , ) ≃ CC( ,  ).

Summary
We summarise the results of this section in the form that we will use later on.

Corollary 3.19. Let be a group and assume we have a short exact sequence
Let be a set of generators for = ⟨ ⟩ and let  be a family of proper subgroups. Furthermore, assume that for all ∈ , one of the following holds: 1. either contains the kernel = ker , or 2. contains ⧵ .
Then there is a homotopy equivalence Proof. We stick with the notation and standing assumption defined in the paragraph before Section 3.2.1. If ∈  , it cannot contain . Hence, all such must contain the set ⧵ of elements from that are not contained in the kernel. It follows that for any 1 , … , ∈  , one has ( 1 ∩ ⋯ ∩ ) = . On the other hand, for every ∈  , our assumption implies that ⊆ . Hence,  is strongly divided by and the claim follows from Theorem 3.18. □

THE BASE CASES: BUILDINGS AND RELATIVE FREE FACTOR COMPLEXES
In this section, we study complexes of parabolic subgroups associated to two particular families of (relative) automorphism groups: The first one is GL (ℤ) (Section 4.1), the second one is given by so-called Fouxe-Rabinovitch groups (Section 4.2). On the one hand, these are special cases of the complexes we will consider in Section 6, on the other hand, they play a distinguished role because they appear as base cases of the inductive argument that we will use there. We show that in both situations, the complexes one obtains are spherical, but the methods for the two cases are quite different. In the first one, the result follows without much effort from the Solomon-Tits theorem while in the second one, we have to generalise the work of [10] to the 'relative' setting considered here.

4.1
The building associated to (ℤ) and the Solomon-Tits theorem The building associated to GL (ℚ) is the order complex of the poset  of proper (that is, non-trivial and not equal to ℚ ) subspaces of ℚ , ordered by inclusion. This is a special case of a Tits building and a lot can be said about the structure of these simplicial complexes -we refer the reader to [2] for further details. However, the only non-trivial result about them that we need for this article is the following special case of the Solomon-Tits theorem.
It is well-known that this building can equivalently be described as the coset complex of GL (ℚ) with respect to the family of maximal standard parabolic subgroups. We will now show that it can also be described as a coset complex of GL (ℤ) = Out(ℤ ), an outer automorphism group of a RAAG.
A subgroup ⩽ ℤ is called a direct summand if there is ⩽ ℤ such that ℤ = ⊕ . We say that a direct summand is proper if it is neither trivial nor equal to ℤ . Let  be the poset of all proper direct summands of ℤ , ordered by inclusion. The group GL (ℤ) acts naturally on .
Fix a basis { 1 , … , } of ℤ and for all 1 ⩽ ⩽ − 1, set ∶= ⟨ 1 , … , ⟩. Note that ∈  for all and define ∶= Stab GL (ℤ) ( ) to be the stabiliser of under the action of GL (ℤ) on . We define the set of maximal standard parabolic subgroups of GL (ℤ) as Remark 4.2. We called the elements of  the maximal standard parabolic subgroups of GL (ℤ) to match the usual convention where an arbitrary parabolic subgroup is defined as the conjugate of a standard one. As we will, however, not work with non-standard parabolic subgroups in this article, we leave out this adjective from now on.
In terms of matrices, the maximal parabolic subgroups can be written in the form  Proof. Each ∈  is isomorphic to ℤ for an integer ∶= rk( ) ∈ {1, … , − 1}, the rank of . Furthermore, if ⩽ in , we have rk( ) ⩽ rk( ) with equality if and only if and are equal. It follows that the maximal simplices of Δ() are given by chains 1 ⩽ ⋯ ⩽ −1 , where rk( ) = .
The group GL (ℤ) acts transitively on the set of all such chains and preserves the rank of each summand. Hence, the facet 1 ⩽ ⋯ ⩽ −1 is a fundamental domain for this action and Proposition 3.9 implies that the order complex of  is GL (ℤ)-equivariantly isomorphic to CC(GL (ℤ), ).
On the other hand, there is a poset map ∶  →  defined by sending to ∩ ℤ . This is a GL (ℤ)-equivariant isomorphism whose inverse is given by sending ⩽ ℤ to its ℚ-span ⟨ ⟩ ℚ (see, for example, [18, Corollary 2.5]). □

Relative free factor complexes
The aim of this section is to generalise [10, Theorem A] which states that the complex of free factors of the free group is homotopy equivalent to a wedge of ( − 2)-spheres. We want to extend this result to certain complexes of free factors of a free product = * 1 * ⋯ * . After adapting the definitions to this setting, the proofs of [10] largely go through without major changes. We still include most of them here in order to make this section as self-contained as possible.

4.2.1
Relative automorphism groups and relative Outer space

Relative automorphism groups
Let be a countable group. We will often use capital letters for elements from the outer automorphism group of and lowercase letters for the corresponding representatives from the automorphism group of ; that is, for Φ ∈ Out( ), we write Φ = [ ], where ∈ Aut( ). Let Φ be an outer automorphism of a group and ⩽ a subgroup. Then Φ stabilises or is invariant under Φ if there exists a representative ∈ Φ such that ( ) = . We say that Φ acts trivially on if there is ∈ Φ restricting to the identity on . If  and  are families of subgroups of , the relative outer automorphism group Out( ; ,  ) is the subgroup of Out( ) consisting of all elements stabilising each ∈  and acting trivially on each ∈ . If  or  are given by the empty set, we also write Out( ;  ) or Out( ; ) for this group.
If ⩽ Out( ) is a subgroup of the outer automorphism group of and ⩽ , we also write Stab ( ) for the subgroup of consisting of all elements that stabilise . In the case where is equal to

Free splittings
A free splitting of is a non-trivial, minimal, simplicial -tree with finitely many edge orbits and trivial edge stabilisers. The vertex group system of a free splitting is the (finite) set of conjugacy classes of its vertex stabilisers. Two free splittings and ′ are equivalent if they are equivariantly isomorphic. We say that ′ collapses to if there is a collapse map ′ → which collapses an -invariant set of edges. The poset of free splittings  is given by the set of all equivalence classes of free splittings of , where ⩽ ′ if ′ collapses to . The free splitting complex is the order complex Δ( ) of the poset of free splittings.

Fouxe-Rabinovitch groups and relative Outer space
Let be a finitely generated group that splits as a free product where denotes the free group on generators and + ⩾ 2. Define  ∶= { 1 , … , } and ∶= Out( ;  ). The group is also called a Fouxe-Rabinovitch group because of the work of Fouxe-Rabinovitch on automorphism groups of free products [21].
In [24], Guirardel and Levitt define a topological space called relative Outer space for such groups. This space contains a spine, which is denoted by = ( , ). This spine is (the order complex of) the subposet of  consisting of all free splittings whose vertex group system is equal to the set of conjugacy classes of elements from . The poset is contractible and acts co-compactly on it.

4.2.2
Parabolic subgroups and relative free factor complexes
A free factor of is a subgroup ⩽ such that splits as a free product = * . There is a natural partial order on the set of conjugacy classes of free factors of given by  Remark 4.5. If = 0, the poset  consists of all conjugacy classes of proper free factors of , so we recover the free factor complex of . More generally, the free factor complex of relative to  is a subcomplex of the complex of free factor systems of relative to  as defined by Handel and Mosher [26]. The ordering ⊑ of free factor systems defined there restricts to the ordering on  for free factor systems having only one component. Our definition, however, differs from the one used by Guirardel and Horbez, for example, in [22]; in their definition, a proper free factor ⩽ is relative to  if = * , where for all , For studying geometric questions, the definition of the free factor complex and similar complexes is often adapted such that it becomes connected for low as well. This is not the case for the definition used in this article, where the free factor complex associated to Out( 2 ) is a disjoint union of points. [26,Lemma 2.11] implies that the elements of  are conjugacy classes of groups of the form

Corank
where ∈ and is a free group with 1 ⩽ rk( ) ⩽ − 1. Furthermore, we can write as a free product = * , where is a free group of rank − rk( ). The rank of is an invariant of the conjugacy class [ ] (see [26,Section 2.3]). It is called the corank of [ ] and will be denoted by crk[ ].
We study these relative free factor complexes because they can also be described as coset complexes of parabolic subgroups: Let Every is a free factor of because for all , we have = * ⟨ +1 , … , ⟩. We set ∶= Stab ( ) and define the set of maximal standard parabolic subgroups of as As in the case of GL (ℤ), we will usually leave out the adjective 'standard' (see Remark 4.2). Proposition 4.6. The free factor complex of relative to  is -equivariantly isomorphic to the coset complex CC( , ). . Consequently, the simplices of Δ( ) are given by chains of the form

Proof. If
We claim that for each such chain, there exists Φ ∈ with [ ( )] = [ ] for all . To see this, first observe that sending each to a conjugate of itself and fixing all the other generators defines an automorphism of that represents an element in . Hence, we can assume that 1 * ⋯ * ⩽ 1 . Now choose representatives such that ⩽ +1 for all . To use induction, assume that there is Φ ′ ∈ such that for some , we have ′ ( ) = for all 0 ⩽ ⩽ -this is true for = 0, where we define 0 = 0 and 0 = 0 = 1 * ⋯ * . By assumption, ′ ( ) = ⩽ +1 , so [26, Lemma 2.11] implies that = ′ ( ) * * , where +1 = ′ ( ) * and and are free groups of rank ( +1 − ) and ( − +1 ), respectively. On the other hand, the group also decomposes as a free product BRÜCK This allows us to define an automorphism of which agrees with ′ on , maps ⟨ +1 , … , +1 ⟩ isomorphically to and ⟨ +1 +1 , … , ⟩ to . As agrees with ′ on , we know that [ ( )] = [ ] for all ⩽ and that acts by conjugation on each , that is, [ ] ∈ . Furthermore, we have By induction, this proves the claim. On the other hand, for each [ ] ∈ , the chain forms a facet in Δ( ). Hence, every facet of Δ( ) can be written in this form. It follows that the natural action of on Δ( ) has a fundamental domain given by the simplex The result now follows from Proposition 3.9. □ Note that the corank played in this proof the same role as the dimension and rank did in the proof of Proposition 4.3.

4.2.3
The associated complex of free splittings To study the connectivity properties of relative free factor complexes, we will use yet another description of them; namely, we will show in this subsection that they are homotopy equivalent to certain posets of free splittings. Let ∶= ( , ) be the spine of Outer space of relative to . Taking the quotient by the action of , each free splitting ∈ can equivalently be seen as a marked graph of groups . The edge groups of are trivial and for all 1 ⩽ ⩽ , there is exactly one vertex group which is conjugate to . All the other vertex groups are trivial. The marking is an isomorphism 1 ( ) → that is well-defined up to composition with inner automorphisms. Using this description, the action of on is given by changing the marking. The underlying graph of is finite, has fundamental group of rank and all of its vertices with valence one have non-trivial vertex group.  For the graph associated to ∈ as above, there is a natural labelling ∶ {1, … , } → ( ) of given by defining ( ) as the vertex with vertex group conjugate to . It follows that ( , ) is a core graph. If is a connected subgraph of that contains all the vertices with non-trivial vertex group, then there is an induced structure of a marked graph of groups on . We define the fundamental group ( ) as the fundamental group of this graph of groups. It is a subgroup of that is well-defined up to conjugacy and has the form where ∈ and is a free group with rank equal to the rank of 1 ( ).
Definition 4.8. Let ∈ , let be the associated graph of groups and ( , ) the underlying labelled graph. Let ⩽ be a subgroup of . We say that has a subgraph with fundamental If such a subgraph exists, there is also a unique core subgraph of ( , ) with fundamental group [ ] which will be denoted by | . We then also say that | is a subgraph of .

Notation
To simplify notation, we will from now onwards not distinguish between a free splitting and the corresponding graph of groups. For example, we will talk about '(core) subgraphs of ' and mean (core) subgraphs of the corresponding labelled graph ( , ). Instead we will use the letter for elements in = ( , ) and the letter for free splittings that have vertex group system different than . If ∈ and is a subgraph, let ∕ denote the free splitting obtained by collapsing .  This proposition can be shown as [10,Theorem 5.8]; there, only the case where is a free group is considered, but the proof generalises to the present situation without any major changes.
In what follows, we provide an outline of the main steps. We start with the case = 0. Let  be the Outer space of relative to {[ ]} as defined in [24]. It can be seen as a subspace of the space of all non-trivial metric simplicial -trees. In [23], Guirardel and Levitt show that its closure in this space is contractible. To do so, they use Skora folding paths to define a map ∶ × [0, ∞] →. The map depends on the choice of a 'base point' 0 ∈  and is defined such that for all , one has ( , 0) = , whereas ( , ∞) is contained in a contractible subspace (a closed simplex of) containing 0 . In [10,Lemma 5.5], it is shown that for an appropriate choice of 0 (namely, for a tree in ( ∶ 0 , … , ) with a minimal number of edge orbits), the map restricts to a continuous map on ‖ ( ∶ 0 , … , )‖. There, the argument is formulated for the case where a free group, but it applies verbatim in our setting as all the results in [23] are formulated in this more general situation anyway.
For > 0, assume that by induction, we know that the posets are contractible. The poset ( , 1 ⋯ , ∶ 0 , … , ) is the union of −1 and . Furthermore, an element ∈ −1 collapses to some ′ ∈ if and only if ∈ −1, . It follows that Each simplex in the order complex Δ( 1 ( )) is given by a sequence 1 , … , , where every is a free splitting in  1 that collapses to +1 . It follows that the vertex groups of these splittings form a chain ( 1 ) ⩽ ⋯ ⩽ ( ) such that [ ] ⩽ ( ) for all . Hence, the simplex is contained in the order complex of ( , ( 1 ), … , ( ) ∶ ). Consequently, the realisation ‖ 1 ( )‖ can be written as a union By the arguments above, all of these sets are contractible. Also, one has This implies that finite intersections of the sets appearing on the righthand side of Equation (1) are contractible. By the nerve lemma (see [4,Theorem 10.6]), ‖ 1 ( )‖ is homotopy equivalent to the nerve of this covering. This is contractible as all of these sets intersect non-trivially (they all contain ‖ ( ∶ )‖). □ Proposition 4.11. There is a homotopy equivalence  1 ≃  .
Proof. Assigning to each splitting ∈  1 the conjugacy class ( ) of its non-trivial vertex stabiliser defines a poset map ∶  1 →  . As there is a natural isomorphism of the order complexes Δ( ) ≅ Δ( ), we will interpret as an order-inverting map ∶  1 →  . Now for any ∈  , the fibre −1 ( ⩾ ) is equal to the poset  1 ( ) which is contractible by Proposition 4.10. The claim follows from Lemma 2.1. □

Homotopy type of relative free factor complexes
To study the homotopy type of  1 , we 'thicken it up' by elements from = ( , ), the spine of Outer space of relative to .

Definition 4.12.
Let be the subposet of the product ×  1 consisting of all pairs ( , ) such that = ∕ is obtained from by collapsing a proper core subgraph and let 1 ∶ → and 2 ∶ →  1 be the natural projection maps.
By analysing the maps 1 and 2 , we now want to show that  is spherical. This closely follows [10, Section 7]. We first deformation retract the fibres of 2 to a simpler subposet. Lemma 4.13. For all ∈  1 , the fibre −1 2 ( 1 ⩾ ) deformation retracts to −1 2 ( ).
) is a well-defined, monotone poset map restricting to the identity on −1 2 ( ). It follows from Corollary 2.4 that this defines a deformation retraction. □ Hence, instead of studying arbitrary fibres, it suffices to consider the preimages of single vertices.
Every element in −1 2 ( ) is given by a pair ( , ) such that ∶= | is a subgraph of and = ∕ . Forgetting the (constant) second coordinate, we can interpret these as elements of the Outer space of relative to . Let be the subspace of this Outer space that is given by all open simplices containing an element of −1 2 ( ). Then −1 2 ( ) is a deformation retract of . In [10, Propositions 7.3.1 and 7.2], Skora folds are used to show that is contractible. The proof in [10] is formulated for the case where is free, but it applies here as well because it only uses the ideas of Guirdel and Levitt [23], which hold true in the generality needed in our setting. □ In particular, these fibres are all contractible, so by Lemma 2.1, we have the following.
We now turn to 1 and show that its fibres are highly connected as well.  Proof. Each element of −1 1 ( ⩽ ) consists of a pair ( ′ , ′ ), where ′ ⩽ in and ′ ∈  1 is obtained from ′ by collapsing a proper core subgraph ′ . As ′ is obtained from by collapsing a forest, ∶= ′ ( ′ )| is a subgraph of .
For every proper core subgraph of , the pair ( , ∕ ) forms an element of −1 1 ( ). Also, if and ′ are proper core subgraphs of , one has ∕ ⩾ ∕ ′ in  1 if and only if ⩽ ′ in C( , ). Hence, the fibre −1 We postpone the proof of this result until Section 4.2.6 and first note the following corollary. The main result of this section, which was stated as Theorem D in the introduction, is now an easy consequence of the last corollaries.

Cohen-Macaulayness
The relative formulations allow us to deduce that  or equivalently CC( , ) is even Cohen-Macaulay. As above, each can be written in the form where is a free group of rank − crk[ ]. Using malnormality of free factors, it follows that two subgroups of a free factor of are conjugate in if and only they are conjugate in (see [26,Lemma 2.1]).It follows that there are isomorphisms The result now follows from Lemma 2.5 and Theorem 4.20.  The following lemma allows us to replace C( , ) with X( , ). This bigger poset will be easier to handle for the inductive arguments that we want to use. Proof. By restricting the labelling, every ∈ X( , ) can be seen as a labelled graph ( , ). It contains contains a unique maximal core subgraph (̊, ). Also, if 1 ⩽ 2 in X( , ), one has (̊1, ) ⊆ (̊2, ). Hence, sending ( , ) to (̊, ) defines a poset map ∶ X( , ) → C( , ) that restricts to the identity on C( , ). The claim now follows from Corollary 2.4. □ An edge ∈ ( ) is called separating if − is disconnected; in particular, we consider edges adjacent to vertices of valence one to be separating. Then X( , ) ⧵ ≃ X( ∕ , ). Furthermore, if is separating, then is empty, so X( , ) ≃ X( ∕ , ).
Proof. Whenever ∈ X( , ), the edges in ( ) ⧵ { } form a connected subgraph of ∕ that will be denoted by ∕ . It contains all labelled vertices of ( ∕ , ) and has non-trivial fundamental group.
If is not in , then either is not adjacent to and hence 1 ( ∕ ) ≅ 1 ( ), or 1 ( ∕ ) ⩽ In either case, 1 ( ∕ ) is a proper subgroup of 1 ( ∕ ) ≅ 1 ( ). Consequently, we get a poset map On the other hand, if ∈ X( ∕ , ) contains the vertex to which was collapsed, it is easy to see that ∪ { } is an element of X( , ) ⧵ . This allows us to define a poset map , else.
One has g• ( ) ⊇ and •g( ) = , so using Lemma 2.3, these two posets are homotopy equivalent, which proves the first part of the statement. For the second part, note that if is separating and ∈ X( , ) such that ∪ { } is connected, then either is contained in or has valence one in ∪ { }. In either case, we have 1 ( ∪ { }) = 1 ( ) ≠ 1 ( ). □ To prove the following result, we apply an argument similar to the one used in [45, Proposition 2.2]. (Note that in [45], being -spherical is only defined for -dimensional posets.) Proof. If ∈ ( ) is separating, then by Lemma 4.24, we have X( , ) ≃ X( ∕ , ). As ∕ has one edge less than , we can apply induction to assume that does not have any separating edges.
We do induction on and start with the case = 2. By Lemma 4.23, it suffices to show that C( , ) is homotopy equivalent to a wedge of 0-spheres, that is, a disjoint union of points. To see this, let ∈ C( , ). As 1 < 1 ( ) < 1 ( ), the fundamental group of is infinite cyclic. Let ∈ be an edge of . We distinguish between the two cases where is non-separating or separating in . If is non-separating, then − has trivial fundamental group while if is separating, − has two connected components both of which either have non-trivial fundamental group or contain at least one labelled vertex. In both cases, no ∈ C( , ) can be contained in − . Hence, the order complex of C( , ) does not contain any simplex of dimension greater than zero which proves the claim. Now let > 2. If every edge of is a loop, is a rose with petals and every proper non-empty subset of ( ) forms an element of X( , ). In this case, the order complex of X( , ) is given by the set of all proper faces of a simplex of dimension − 1 whose vertices are in one-to-one correspondence with the edges of and hence is homotopy equivalent to an ( − 2)-sphere.
So, assume that has an edge that is not a loop. As we assumed that is non-separating, − is a connected graph having the same number of vertices as and one edge less. This implies that rk( 1 ( − )) = − 1. Collapsing separating edges and using Lemma 4.24, we see that X( − , ) ≃ X( ′ , ) where ′ has the same rank as − , at most as many edges and no separating edges. Hence, X( − , ) ≃ X( ′ , ) is by induction homotopy equivalent to a wedge of ( − 3)-

RELATIVE AUTOMORPHISM GROUPS OF RAAGs
In this section, we examine relative automorphism groups of right-angled Artin groups. These groups were studied in detail in [19] and many of the results here are either taken from the work of Day-Wade or build on their ideas. For an overview about other literature on relative automorphism groups, see [19,Section 6.1]. In this article, such relative automorphism groups occur in two ways: On the one hand, they arise as the images and kernels of restriction and projection homomorphisms, which in turn play an important role for the inductive procedure of Day-Wade; on the other hand, the parabolic subgroups we will define in Section 6 are themselves relative automorphism groups of RAAGs. For the purpose of this text, this section mostly serves as a toolbox for the inductive proof of Theorem A in Section 6. Its main goals are to collect all the results from [19] that we will need afterwards, to adapt them to our purposes and, maybe most importantly, to set up the language we will use later on.

Standing assumption
From now on, all graphs that we consider will be finite and simplicial, that is, without loops or multiple edges. To emphasise this difference to Section 4, they will be denoted by Greek letters.

Subgraphs, links and stars
In contrast to Section 4, if we talk about a subgraph Δ of a graph Γ, we will from now on always mean a full subgraph, that is, if two vertices , ∈ (Δ) are connected by an edge in Γ, they are connected in Δ as well. A full subgraph of Γ can also be seen as a subset of the vertex set (Γ); we will often take this point of view, identify Δ with (Δ) and write Δ ⊆ Γ or Δ ⊂ Γ if we want to emphasise that Δ is a proper subgraph of Γ. Given a vertex ∈ (Γ), the link lk( ) of is the subgraph of Γ consisting of all the vertices that are adjacent to . The star st( ) of is the subgraph of Γ with vertex set { } ∪ lk( ). We also write lk Γ ( ) or st Γ ( ) if we want to distinguish between links and stars in different graphs.

RAAGs and special subgroups
Given a graph Γ, the associated right-angled Artin group -abbreviated as RAAG -Γ is defined to be the group generated by the set (Γ) subject to the relations [ , ] = 1 for all , ∈ (Γ) which are adjacent to each other.
Given any subgraph Δ ⊆ Γ, the inclusion (Δ) → (Γ) induces an injective homomorphism Δ ↪ Γ . This allows us to interpret Δ as a subgroup of Γ . Subgroups of this type are called special subgroups of Γ .

The standard ordering and its equivalence classes
There is a so-called standard ordering on the vertex set (Γ) that is the partial pre-order given by ⩽ if and only if lk( ) ⊆ st(

Automorphisms of RAAGs
Let Aut( Γ ) and Out( Γ ) denote the automorphism group and the group of outer autmorphisms of Γ , respectively. By the work of Servatius [40] and Laurence [33], the group Aut( Γ ) is generated by the following automorphisms.
• Graph automorphisms. Any automorphism of the graph Γ gives rise to an automorphism of Γ by permuting the generators of the RAAG. • Inversions. Let ∈ (Γ). The map sending to −1 and fixing all the other generators induces an automorphism of Γ . It is called an inversion and denoted by . • Transvections. Let , ∈ (Γ) with ⩽ . The transvection is the automorphism of Γ induced by sending to and fixing all the other generators. We call the acting letter of .
• Partial conjugations. Let ∈ (Γ) and a union of connected components of Γ ⧵ st( ). The map sending every vertex of to −1 and fixing the remaining generators induces an automorphism of Γ and is called a partial conjugation. We call the acting letter of .
We will use the same notation to denote the images of these automorphisms in Out( Γ ) and call these (outer) automorphisms the Laurence generators of Aut( Γ ) or Out( Γ ), respectively.
The subgroup of Out( Γ ) generated by all inversions, transvections and partial conjugations is denoted by Out 0 ( Γ ). It is called the pure outer automorphism group of Γ and has finite index in Out( Γ ). If Γ is equal to ℤ or , we have that Out 0 ( Γ ) = Out( Γ ).

Generators of relative automorphism groups
Recall that for a group and families of subgroups  and , the group Out( ; ,  ) is defined as the subgroup of Out( ) consisting of all elements stabilising each ∈  and acting trivially on each ∈  (see Section 4.2.1). Given a pair (, ) of families of special subgroups of Γ , we define To prove this, Day and Wade give a description of the Laurence generators contained in such a relative automorphism group. To state it, we first need to set up the terminology developed in their article.

-components and -ordering
Let  be a family of proper special subgroups of Γ . We say that , ∈ ( Γ ) are -adjacent if is adjacent to or if there is some Δ ∈  such that , ∈ Δ. A subgraph Δ ⊆ Γ is -connected if for all , ∈ Δ, there is a sequence of vertices in Δ which starts with , ends with and such that each of its vertices is -adjacent to the next one. A maximal -connected subgraph of Γ is called a -component.
The -ordering ⩽  on (Γ) is the partial pre-order defined by saying that ⩽  if and only if ⩽ and for all Δ ∈ , if ∈ Δ, one has ∈ Δ. The equivalence relation of this pre-order is denoted by ∼  , its equivalence classes by [⋅]  .
Note that in the case where  = ∅, a -component of Γ is just a connected component and the -ordering is the standard ordering on (Γ).
For ∈ (Γ), let  ∶= { Δ ∈  | ∉ Δ}. It is easy to see that every  -component of Γ ⧵ st( ) is a union of connected components of Γ ⧵ st( ). Suppose that  is a family of special subgroups of Γ . The power set of , denoted by (), is defined as the set of all special subgroups Δ ⩽ Γ which are contained in some element of .

 ) if and only if is a union of components of Γ ⧵ st( ).
Note that it imposes no great restriction to assume that the power set of  be contained in  because for any families  and  of special subgroups, one has Out 0 ( Γ ; ,  ) = Out 0 ( Γ ;  ∪ (),  ) (see [19,Lemma 3.8]).
The next result is the key ingredient for the proof of [19, Lemma 5.2]. We include it here because it will allow us a more convenient description of the parabolic subgroups that we will study later on.

Restriction and projection homomorphisms
Let be a subgroup of Out( Γ ). If the special subgroup Δ ⩽ Γ is stabilised by , there is a restriction homomorphism where Δ (Φ) is the outer automorphism given by taking a representative ∈ Φ that sends Δ to itself and restricting it to Δ . If the normal subgroup ⟨⟨ Δ ⟩⟩ generated by Δ is stabilised by , there is a projection homomorphism which is induced by the quotient map Γ → Γ ∕⟨⟨ Δ ⟩⟩ ≅ Γ⧵Δ .
Restriction and projection maps were first defined in [13] and have since become an important tool for studying automorphism groups of RAAGs via inductive arguments.

Generators of image and kernel
Day-Wade obtained a complete description of the image and kernel of restriction homomorphisms. Again let  and  be families of special subgroups of Γ . We say that  is saturated with respect to (, ), if it contains every proper special subgroup stabilised by Out 0 ( Γ ; ,  ). Given a special subgroup Δ ⩽ Γ , set We define  Δ analogously. and kernel equal to It is not hard to see that both restriction and projection maps send each Laurence generator either to the identity or to a Laurence generator of the same type. For the proof of Theorem 5.4, Day-Wade show that for restriction maps, a converse of this is true as well: Every Laurence generator in im Δ is given as the restriction of a Laurence generator of Out 0 ( Γ ; ,  ).
In general, image and kernel of projection homomorphisms are more difficult to describe. However, we will only need to consider them in a special case: The centre ( Γ ) of Γ is generated by all vertices ∈ (Γ) such that st( ) = Γ. If ( Γ ) is non-trivial, these vertices form an abelian equivalence class ∶= [ ] and Γ can be written as a join Γ = * Δ, where Δ = Γ ⧵ . If we have a graph of this form, the centre ( Γ ) = is a normal subgroup which is stabilised by all of Out( Γ ). Hence, there is a projection map The image of this projection map can be described very similar to the one of a restriction map. In fact, the situation in this special case is even easier as we do not even need to assume any kind of saturation for our families of special subgroups.
Lemma 5.5. Assume that Γ can be decomposed as a join Γ = * Δ where is a complete graph. Let  and  be any two families of special subgroups of Γ and let ∈ . The projection homomorphism Δ ∶ Out 0 ( Γ ; ,  ) → Out( Δ ) has image equal to im Δ = Out 0 ( Δ ;  Δ ,  Δ ).

Proof. The inclusion '⊆' follows immediately from the definitions.
For the other inclusion, we start by defining ∶=  ∪ () as the union of  and the power set of . As observed above, we have ∶= Out 0 ( Γ ; ,  ) = Out 0 ( Γ ;,  ).
(2) By Theorem 5.1, we know that Δ is generated by the inversions, transvections and partial conjugations it contains. Hence, it suffices to find a preimage under Δ for each of those generators. Combining Equation (2)  In what follows, we will use this description to construct the preimages one generator at a time.
The inversion is contained in Δ if and only if ∈ Δ and there is no Δ ′ ∈  Δ such that ∈ Δ ′ . However, this implies that there is no Δ ′ ∈  with ∈ Δ ′ , so the inversion at is an element of . It will be denoted bȳand gets mapped to under Δ .
Again using Lemma 5.2, the partial conjugation is contained in Δ if and only if ∈ Δ and is a union of Δ -components of Δ ⧵ st( ). We claim that every Δ -component of Δ ⧵ st Δ ( ) is also a -component of Γ ⧵ st Γ ( ). To see this, first recall that each element of is connected to every vertex of Γ, so Γ ⧵ st Γ ( ) = Δ ⧵ st Δ ( ). Furthermore, it follows right from the definitions that two vertices , ∈ Δ ⧵ st Δ ( ) are Δ -adjacent in Δ ⧵ st Δ ( ) if and only if they are-adjacent in Γ ⧵ st Γ ( ). The claim follows and implies that the partial conjugation of by defines an element of . As above, it will be denoted bȳand we note that it is a preimage of . □

Standing assumptions and notation
From now on and until the end of Section 5, let ∶= Out 0 ( Γ ; ,  ), where  and  are families of special subgroups of Γ such that  is saturated with respect to (, ); note that saturation implies that () ⊆ . Set ⪯∶=⩽  to be the -ordering on (Γ).
Remark 5.6. Given an arbitrary relative automorphism group, there might be several ways of 'representing' this group by families of subgroups that are stabilised or acted trivially upon, that is, we might have Out 0 ( Γ ;  1 ,  1 ) = Out 0 ( Γ ;  2 ,  2 ) with ( 1 ,  1 ) ≠ ( 2 ,  2 ). However, if in this situation, we have both ( 1 ) ⊆  1 and ( 2 ) ⊆  2 , the orderings ⩽  1 and ⩽  2 agree: By Lemma 5.2, for every , ∈ (Γ), there is a chain of equivalences In particular, the ordering ⩽  of ( Γ ) where  is saturated with respect to (, ) is an invariant of the group Out 0 ( Γ ; ,  ); it depends on the transvections contained in this group but not on any other choices.
As mentioned above, a restriction homomorphism maps every transvection that is not contained in its kernel to a transvection of the same type. The consequences for the relative ordering in image and kernel are as follows.
Lemma 5.7. Let Δ ∈  be a special subgroup that is stabilised by and let Δ denote the corresponding restriction homomorphism. If we write im Δ = Out 0 ( Δ ;  im ,  im ) and ker Δ = Out 0 ( Γ ;  ker ,  ker ) with  im and  ker saturated with respect to ( im ,  im ) and ( ker ,  ker ), respectively, the following holds true.
For the second point, we have ⩽  ker if and only if ∈ ker Δ . This is the case if and only if is contained in and acts trivially on Δ . The claim now follows from Lemmas 5.2 and 5.3. the family  ∪ { Δ } is not necessarily saturated with respect to ( ∪ { Δ }, ) and its image under Δ is more difficult to describe. However, the parabolic subgroups we will consider in Section 6 are exactly of this form. The next two lemmas show that in special cases, we can describe their images under Δ without passing to saturated pairs. Lemma 5.8. Assume that stabilises a special subgroup Δ ⩽ Γ and let Δ ∶ → Out( Δ ) denote the corresponding restriction homomorphism. Take Θ ⊂ Γ. Then, Proof. The first point becomes tautological after spelling out the definitions.
Proof. The stabiliser Stab ( Θ ) is the same as the relative automorphism group Out 0 ( Γ ;  ∪ { Θ },  ). By Lemma 5.5, the image of this group is equal to On the other hand, we have im Δ = Out 0 ( Δ ;  Δ ,  Δ ), so the right-hand side of Equation (3) is also equal to Stab im Δ ( Θ∩Δ ) and the claim follows. □

Restrictions to conical subgroups
In this section, we define a family of special subgroups that will play an important role in our inductive arguments later on and study some properties of these special subgroups. For a vertex ∈ (Γ), define the following subgraphs of Γ: where ≺ if ⪯ and ≁  . We define ⪰ ∶= Γ ⪰ and ≻ ∶= Γ ≻ as the special subgroups of Γ corresponding to these subgraphs. Note that these special subgroups only depend on the ∼  -equivalence class of , that is, if ∼  , we have ⪰ = ⪰ . In the 'absolute setting' where  and  are trivial and ⪯ is equal to the standard ordering of (Γ), these special subgroups appear as admissible subgroups in the work of Duncan and Remeslennikov [20]. We will also refer to them as conical subgroups of Γ as they are generated by elements corresponding to an upwards-closed cone in the Hasse diagram of the partial order that ⪯ induces on the equivalence classes of ∼  (see Figure 2). The elements of Out 0 ( Γ ) are characterised among all elements of Out( Γ ) by the property that they stabilise these special subgroups. Namely, the following holds true.
In particular, each of these special subgroups is stabilised by all of Out 0 ( Γ ). We will need a relative version of this statement.

Lemma 5.11. Let , ∈ (Γ), let be a union of  -components of Γ ⧵ st( ) and let
∈ denote the corresponding partial conjugation. If  , the partial conjugation acts trivially on ⪰ .
Proof. As is not smaller than with respect to ⪯, there either is an element in lk( ) which is not contained in st( ) or there is Δ ∈  such that Δ contains but does not contain . We claim that in both cases, Γ ⪰ intersects at most one  -component of Γ ⧵ st( ). Indeed, if there is ∈ lk( ) ⧵ st( ), one has ∈ st( ) ⧵ st( ) for all ∈ Γ ⪰ . Hence, all elements of Γ ⪰ are adjacent to and Γ ⪰ ⧵ st( ) is contained in a single  -component of Γ ⧵ st( ). If on the other hand for some Δ ∈ , one has ∈ Δ, it follows that ∈ Δ for all ∈ Γ ⪰ . Now if ∉ Δ, this implies that all elements of Γ ⪰ are  -adjacent, so they in particular lie in the same  -component. Either way, Lemma 5.3 implies that acts trivially on ⪰ . □ Proposition 5.12. For every vertex ∈ (Γ), the special subgroup ⪰ is stabilised by every element from .
Proof. As is generated by the inversions, transvections and partial conjugation it contains, it suffices to prove the statement for each such element. As above, this can be done using Lemmas 5. 2 and 5.3. For inversions, there is nothing to show as they always stabilise every special subgroup. If we have a transvection ∈ , we must have ⪯ . The set Γ ⪰ is upwards-closed with respect to ⪯, hence ∈ Γ ⪰ implies ∈ Γ ⪰ . It follows that stabilises ⪰ . Given a partial conjugation ∈ , we either have  , in which case Lemma 5.11 implies that even acts trivially on ⪰ , or we have ∈ Γ ⪰ which implies that stabilises ⪰ . □ A consequence of this is that for every equivalence class [ ]  of vertices of Γ, we have a restriction map ⪰ = ⪰ ∶ → Out 0 ( ⪰ ).
These maps are crucial for the line of argument in the following section. We will study some of their properties in Lemma 6.6.

A SPHERICAL COMPLEX FOR ( )
In this section, we define maximal parabolic subgroups of Out 0 ( Γ ) in the general case. We then prove Theorem A which states that the coset complex associated to these parabolic subgroups is homotopy equivalent to a wedge of spheres.

Notation and standing assumptions
As before, let Γ be a graph,  and  families of special subgroups of Γ such that  is saturated with respect to (, ), define ∶= Out 0 ( Γ ; ,  ) and set ⪯∶=⩽  to be the -ordering on (Γ). Let  denote the set of ∼  -equivalence classes of vertices of Γ. Proof. Again, we use Lemmas 5.2 and 5.3: As all vertices of [ ]  are equivalent with respect to ⩽  , the transvection 1 is an element of . However, this transvection does not stabilise Δ because

Rank and maximal parabolic subgroups
Definition 6.3. We define the set of maximal standard parabolic subgroups of as the union The reader might at this point want to verify that for the graph Γ depicted in Figure 2, one has |(Out 0 ( Γ ))| = 4. The term 'maximal' parabolic will become clear in Section 8.1 where we will define and study parabolic subgroups of lower rank. As before, we will usually leave out the adjective 'standard' (see Remark 4.2). Remark 6.4. We note the following properties of ( ) and rk( ).
1. rk( ) = |( )|. We will also give an alternative interpretation of rk( ) in Section 8.3. 2. By Lemma 6.2, every element of ( ) is a proper subgroup of . 3. Following Remark 5.6, the definition of parabolic subgroups depends on the ordering chosen for each equivalence class, but not on the pair (, ) we chose to represent . 4. If is equal to GL (ℤ) or a Fouxe-Rabinovitch group, we recover the definitions of parabolic subgroups in these groups as defined in Sections 4.1 and 4.2. Furthermore, rk(GL (ℤ)) = rk(Out( )) = − 1.
Note that it is possible that there is no -equivalence class of size bigger than one. In this case, the rank of is zero and ( ) is empty. For further comments on this, see Section 9.

The parabolic sieve
In this subsection, we explain the idea of the inductive argument that we will use to show sphericity of the coset complexes CC( , ( )).

Outline of proof
Whenever Δ ⊂ Γ is stabilised by , the restriction map Δ gives rise to a short exact sequence and by Theorem 5.4, both and are relative automorphism groups of RAAGs again. Using the considerations of Section 5, we will show that for the correct choice of Δ, every ∈ ( ) satisfies the following dichotomy: Either Δ ( ) is contained in ( ) or ∩ forms an element of ( ). Applying a restriction homomorphism hence has the effect of a sieve on ( ) -some of the parabolic subgroups pass through and form parabolics of the quotient while others remain in the sieve and form parabolics of the subgroup . Now using the results of Section 3, this allows us to describe the homotopy type of CC( , ( )) in terms of the topology of the lower dimensional coset complexes CC( , ( )) and CC( , ( )). This is used for an inductive argument with two phases: We first apply restriction maps to conical subgroups (Section 6.2.1) and then analyse the homotopy type of coset complexes in the conical setting (Section 6.2.2). Concrete examples of this induction will be given in Section 7.
We want to apply Corollary 3.19 to prove this statement. To do so, we have to show that for each ∈ ( ), either ker ⊆ or contains all inversions, transvections and partial conjugations of that are not contained in ker . This is the content of the following lemma.
Lemma 6.6. The restriction map = ⪰ has the following properties.
2. For all ∈ (Γ), the following holds: If Δ ⊆ Γ ⪰ such that the stabiliser Stab ( Δ ) contains all inversions, transvections and partial conjugations of that are not contained in ker .
Proof. By Theorem 5.4, the kernel of consists of all elements from that act trivially on the special subgroup ⪰ . This immediately implies the first claim.
For the second one, we again use Lemmas 5. 2 and 5.3. First note that Stab ( Δ ) contains all inversions of .
Next assume we have a transvection ∈ . If ∉ Γ ⪰ , the transvection is contained in ker . If on the other hand ∉ Δ, the transvection acts trivially on Δ and hence is contained in Stab ( Δ ). Now observe that the assumption that Γ ⪰ ∩ Γ ⪰ ⊆ Δ implies that Δ ∩ Γ ⪰ is equal to Γ ⪰ ∩ Γ ⪰ , a set which is upwards-closed with respect to ⪯. So if ∈ Δ ∩ Γ ⪰ , we also have ∈ Δ ∩ Γ ⪰ . Again it follows that ∈ Stab ( Δ ). Finally, consider a partial conjugation ∈ . If  , Lemma 5.11 implies that is contained in ker . This lemma also show that if  , the partial conjugation acts trivially on ⪰ , and hence is contained in Stab ( Δ ). The only case that remains is that is greater than both and , that is, ∈ Γ ⪰ ∩ Γ ⪰ . As we assumed that Γ ⪰ ∩ Γ ⪰ ⊆ Δ, this implies that ∈ Δ, so again ∈ Stab ( Δ ). □ Proof of Lemma 6.5. Set  ∶= ( ).
Take [ ]  ∈  and = Stab ( Δ ) ∈  [ ]  with Δ = Δ as above. If ⪯ , we have Δ ⊂ Γ ⪰ . Hence by the first point of Lemma 6.6, we know that ker ⊆ . If on the other hand ≺ , one has Γ ⪰ ∩ Γ ⪰ = Γ ⪰ ⊂ Δ. Similarly if and are incomparable, one has Γ ⪰ ∩ Γ ⪰ ⊆ Γ ≻ ⊂ Δ. In both cases, the second point of Lemma 6.6 tells us that contains all inversions, transvections and partial conjugations of which are not contained in ker . From this, it follows that with notation as defined in the paragraph before Section 3.2.1. Corollary 3.19 now shows that there is a homotopy equivalence where  = { ( ) | ∈  ker } and  ∩ ker = { ∩ ker | ∈  ker }.

This finishes the proof. □
For the first phase of our induction, we now use this iteratively in order to obtain the following proposition.

By Theorem 5.4, we can write
where  ker is saturated with respect to the pair ( ker ,  ∪ { ⪰ }). Furthermore, Lemma 5.7 together with the third hypothesis of our induction imply that for all ∈ Θ, either ker acts trivially on ⪰ or Θ ⩾  ker = Θ ⪰ . It follows that CC(ker , (ker )) satisfies the hypotheses of our induction. Again using Theorem 5.4, we can write where  im is saturated with respect to ( im ,  im ) and the elements of  im are the special subgroups generated by the vertices of Δ ∩ Γ ⪰ for some Δ ∈  . This implies that CC(im , (im )) satisfies the first hypotheses of our induction. The third one is an immediate consequence of Lemma 5.7. Now apply induction to these coset complexes. This process ends if we arrive at a case where for all ≻ , the group acts trivially on ⪰ . But then we can set  =  Θ ∪ { ⪰ | ≺ } and for ≠ , ⩽  is only possible if ∈ [ ]  . If = 0, this means that the relative ordering on Γ ⪯0 = Γ is trivial, so ( ) = ∅. If ≠ 0, the group ∶= satisfies all conditions of the claim. □

Coset complexes of conical RAAGs
We now want to deal with the coset complexes CC( , ( )) of conical RAAGs that we obtained in Proposition 6.7. This is why in this subsection, we impose the following assumptions.

Standing assumptions
Until the end of Section 6.2.2, we assume that: 1. there is a vertex ∈ (Γ) such that Γ = Γ ⪰ , that is, every vertex of Γ is greater than or equal to with respect to ⪯; 2. for all ≻ , the group acts trivially on the special subgroup { } ⩽ Γ .
Observe that Item 1 implies that for all Δ ∈ , we have Δ ⊆ Γ ≻ ∶ By Lemmas 5.2 and 5.3, every Δ ⊆ Γ such that stabilises Δ must be upwards-closed with respect to ⪯. Hence, if Δ intersects [ ]  non-trivially, it follows that Δ = Γ. Furthermore, Item 2 implies that for all ≻ , the equivalence class [ ]  is a singleton. Hence, we have In this situation, let

∶= { ∈ (Γ) | ≺ and is adjacent to }.
We define the group of twists by elements in Γ ≻ as the subgroup ⩽ generated by the transvections with ∈ [ ]  and ∈ . Lemma 6.8. is a free abelian group. Furthermore, Γ can be decomposed as a join Γ = * Δ and there is a short exact sequence Proof. If = ∅, the statement is trivial, so we can assume that contains at least one element. By definition, we have ⊆ lk( ) ⧵ [ ]  . As every vertex of Γ is greater than or equal to with respect to ⪯, this implies that is a complete graph and we can write Γ = * Δ.
Using the assumption that acts trivially on { } for all ≻ , Lemmas 5.2 and 5.3 imply that acts trivially on the normal subgroup ⊲ Γ . Consequently, we have a welldefined projection map Δ ∶ → Out( Δ ). By Lemma 5.5, the image of this map is equal to Out 0 ( Δ ;  Δ ,  Δ ).
The ordering ⩽  Δ is just the restriction of ⪯ to Δ, so Lemma 5.9 implies that ( ) = (im Δ ). □ We now distinguish between the case where [ ]  is an abelian and the case where it is a free equivalence class. is homotopy equivalent to the Tits building associated to GL (ℚ).
By assumption, the abelian equivalence class [ ]  contains at least two elements which are adjacent to each other. As every vertex of Γ is greater than or equal to with respect to ⪯, this implies that every vertex of Γ ≻ must be adjacent to . Hence, = Γ ≻ and Δ = [ ]  . As observed above (Equation (5)), every Θ ⊆ Γ with Θ ∈  is entirely contained in Γ ≻ . Consequently, we have  Δ =  Δ = ∅ and Out 0 ( Δ ;  Δ ,  Δ ) = GL (ℤ).
This means that CC( , ( )) ≃ CC(GL (ℤ), (GL (ℤ))) and this coset complex is isomorphic to the Tits building associated to GL (ℚ) by Proposition 4.3. □ In the setting of a free equivalence class, the situation is slightly more complicated: As before, we start by projecting away from , but we then might have to apply further restriction maps. Proof. Again by Lemma 6.9, we have a homotopy equivalence CC( , ( )) ≃ CC(im Δ , (im Δ )), where Δ = Γ ⧵ and im Δ = Out 0 ( Δ ;  Δ ,  Δ ). As noted above, the  Δ -ordering on Δ is just the restriction of ⪯ to Δ; in particular we have [ ]  Δ = [ ]  and Δ = Δ ⪰ .
As no two vertices from [ ]  are adjacent to each other, the link lk Γ ( ) is entirely contained in , so every element of [ ]  forms an isolated vertex of Δ. This implies that Δ decomposes as a disjoint union Δ = [ ]  ⊔ ⨆ Δ , where each Δ is a  Δ -component of Δ. In particular, we have Moreover, for all , the group im Δ stabilises Δ : If Δ contains at least two vertices, this is [19,Lemma 3.13.1] and if Δ is a singleton, the action on Δ is trivial by assumption.
If there is an such that im Δ acts non-trivially on Δ , there is a non-trivial restriction map ∶ im Δ → Out( Δ ). Its kernel can be written as where  ker is saturated with respect to ( ker ,  Δ ∪ { Δ }). One can easily check that each ∈ (im Δ ) contains all the inversions, transvections and partial conjugations not contained in ker : The kernel contains all inversions and transvections from im Δ as well as the partial conjugations that have acting letter contained in [ ]  . The remaining partial conjugations are contained in all of the parabolic subgroups. Hence, by Corollary 3.19, we have a homotopy equivalence CC(im Δ , (im Δ )) ≃ ∅ * CC(ker ,  ∩ ker ) = CC(ker ,  ∩ ker ).
Lemma 5.7 implies that the ordering ⩽  ker agrees with ⪯ on Δ; hence using Lemma 5.8, we obtain (ker ) =  ∩ ker . All the Δ are stabilised by ker , so we can use induction and apply restriction maps until we reach the group Out 0 ( Δ ; { Δ } ). This group is equal to Out( Δ , { Δ } ) and hence a Fouxe-Rabinovitch group. The claim now follows from Proposition 4.6. □ Using the results of Section 4, the last two lemmas can be summarised as follows. Proof. If = 1, the statement is trivial as in this case, the set ( ) =  [ ]  is empty. Hence, the complex CC( , ( )) is the empty set which we consider to be (−1)-spherical (see Section 2.3). Now let ⩾ 2.
If [ ]  is abelian, Lemma 6.10 implies that the coset complex is homotopy equivalent to the Tits building associated to GL (ℚ) which is ( − 2)-spherical by the Solomon-Tits theorem. If on the other hand [ ]  is free, it is by Lemma 6.11 homotopy equivalent to a relative free factor complex which is by Theorem 4.20 ( − 2)-spherical as well. □

Proof of Theorem A
We return to the general situation where Γ is any graph and  and  are any families of special subgroups of Γ such that  is saturated with respect to (, ). Recall that ⪯ denotes the -ordering of (Γ) and  denotes the set of associated ∼  -equivalence classes. The only thing that is left to be done for the proof of Theorem A, which we restate below, is to collect the results obtained in Section 6.2. Theorem 6.13. Let ∶= Out 0 ( Γ ; ,  ). The coset complex CC( , ( )) is homotopy equivalent to a wedge of spheres of dimension rk( ) − 1.
Let [ ]  ∈  . Condition 3 implies that the  -equivalence class of in Γ ⪰ is equal to [ ]  and that all other ∈ Γ ⪰ are greater than with respect to ⩽  . Now Condition 2 implies that for all with >  , the group acts trivially on { } ⩽ ⪰ . Hence, the assumptions of Section 6.2.2 are fulfilled and Corollary 6.12 implies that CC( , ( )) is spherical of dimension |[ ]  | − 2. It follows from Lemma 2.5 that the join of these complexes is spherical of dimension

Consequences for the induction of Day-Wade
The proof of Theorem 6.13 relies on the inductive procedure defined in [19]: The authors there show that for every graph Γ, the group Out 0 ( Γ ) has a subnormal series such that for all , the quotient +1 ∕ is isomorphic to either a free abelian group, to GL (ℤ) or to a Fouxe-Rabinovitch group (see [19,Theorem A]). The methods we use in Section 6.2 provide more detailed information about this inductive procedure which decomposes Out 0 ( Γ ) in terms of short exact sequences related to restriction and projection homomorphisms: We are able to give an explicit description of the restriction and projection maps that one has to use during the induction and of the base cases one obtains this way. In what follows, we will give a summary of these results (see also Figure 3). To simplify notation, we will describe the decomposition of = Out 0 ( Γ ), however, all of this can also be stated in the more general case where is any relative automorphism group of a RAAG.
Step 1: First one iteratively restricts to conical subgroups ⩾ until one is left with relative automorphism groups that act trivially on all of their proper conical subgroups -for this, one needs to apply exactly one restriction map for every (standard) equivalence class of (Γ) and the order in which one applies the corresponding restriction maps does not change the base cases of this first induction step. One of these base cases is given by the intersection of the kernels of all the conical restriction maps; it is the group Out 0 ( Γ ; { ⩾ | ∈ (Γ)} ) which does not contain any inversions or transvections. The other base cases are all of the form Out 0 ( ⩾ ; , { ⩾ | ∈ Γ > } ) for some ∈ (Γ) and some family  of special subgroups of ⩾ . There is exactly one such base case for every equivalence class [ ] of (Γ) and it is generated by all the restrictions to ⩾ of inversions, transvections and partial conjugations of Out 0 ( Γ ) that act trivially on ⩾ for every > .
Step 2: Now for each of these groups, one applies the (possibly trivial) projection map Δ , where Δ ∶= Γ ⩾ ⧵ and is the full subgraph of Γ ⩾ consisting of all those vertices of Γ which are adjacent to and strictly greater than with respect to the standard ordering on (Γ). The kernel of this projection map is given by the free abelian group generated by all twist of elements in [ ] by elements in Γ > . We now have to distinguish two cases: If [ ] is an abelian equivalence class of size ⩾ 2, then the image of Δ is given by Out( [ ] ) ≅ GL (ℤ). If this is not the case, we proceed with Step 3.
Step 3: If [ ] is a free equivalence class, the graph Δ decomposes as a disjoint union Δ = [ ] ⊔ ⨆ Δ , where each Δ is a relative connected component of im( Δ ). One can show that the Δ are precisely the non-empty intersections Δ = (Δ ⧵ [ ]) ∩ Γ , where Γ is a connected component of Γ ⧵ lk( ). We now iteratively apply the restriction maps Δ . This yields two kinds of base cases: The first one is given by the intersection of the kernels of all the Δ and can be described as the Fouxe-Rabinovitch group Out( Δ , { Δ } ). The second one is given by the images of the restriction maps. For each , this is a relative automorphism group of Δ ; as Δ ⊆ Γ > , this group contains no inversions or transvections and is generated by partial conjugations.

The base cases
In summary, our induction yields the following base cases: If we order [ ] as ( , ), the family of maximal parabolic subgroups of the group ∶= Out 0 ( Γ ) is given as and we have rk( ) = , that is, CC( , ( )) is ( − 1)-spherical.
After restricting to these conical subgroups (Step 1 of our induction), we are left with the following base cases. Only the groups of the last item have a non-empty set of parabolic subgroups (each given by the singleton {Stab Out(⟨ , ⟩) (⟨ ⟩)}). All items but the first one describe Fouxe-Rabinovitch groups, so the induction already ends here and we do not have to apply Steps 2 and 3.
(For the second isomorphism, we used that contains Out(⟨ , ⟩) for ≠ .) Each factor in this join is a copy of the free factor complex associated to Out( 2 ) and acts on their join in the obvious way.
In Step 2, we apply for each group of the last item the projection map Δ , where Δ = lk( ). Its kernel is a free abelian group of rank , generated by the twists of the leafs adjacent to . The image of this projection map is the Fouxe-Rabinovitch group  We do not spell out the consequences for all of the induction, but would like to point out the following implication for the ranks of the corresponding automorphism groups and hence the dimensions of the associated coset complexes rk(Out 0 ( Γ )) = { rk(Out 0 ( Γ 1 )) + rk(Out 0 ( Γ 2 )), = ∅ for some ; rk(Out 0 ( Γ 1 )) + rk(Out 0 ( Γ 2 )) + 1, otherwise.
Note that = ∅ is equivalent to saying that the centre ( Γ ) is trivial. A particularly simple instance of this is the situation where Γ = × with , > 1. Then, Out 0 ( Γ ) = Out( ) × Out( ) and the coset complex CC(Out 0 ( Γ ), (Out 0 ( Γ )) ) is isomorphic to the the join of the free factor complexes associated to Out( ) and Out( ).
This allows for example to generalise the example of tree-RAAGs given above to the setting of forests.

Complement graph
For a graph Γ, let Γ denote its complement, that is, the graph with vertex set In particular, one has rk(Out 0 ( Γ )) = rk(Out 0 ( Γ )). This also explains the analogy between the settings of direct and free products considered above.

COHEN-MACAULAYNESS, HIGHER GENERATION AND RANK
In this section, we generalise the results of Section 6: We show that the coset complex of parabolic subgroups of a relative automorphism group of a RAAG is not only spherical, but even Cohen-Macaulay. This is used to determine the degree of generation that families of (possibly nonmaximal) parabolic subgroups provide. We also give an interpretation of the rank in terms of a 'Weyl group' of .

Notation and standing assumptions
As before, let Γ be a graph,  and  families of special subgroups of Γ such that  is saturated with respect to (, ), define ∶= Out 0 ( Γ ; ,  ) and set ⪯∶=⩽  to be the -ordering on (Γ). Let  denote the set of ∼  -equivalence classes of vertices of Γ.

Cohen-Macaulayness
For coset complexes, the Cohen-Macaulay property can be characterised as follows. This allows us to generalise Theorem 6.13 in the following way. Proof. By Theorem 8.1, it suffices to show that for all  ′ ⊆ , the coset complex CC( ,  ′ ) is (| ′ | − 1)-spherical. This can be done following the induction of Section 6.2: We first iteratively apply restriction maps to conical subgroups as in Section 6.2.1. In each step, the parabolic subgroups in  ′ satisfy a dichotomy that allows us to apply Corollary 3.19. We get an analogue of Proposition 6.7: The coset complex CC( ,  ′ ) is homotopy equivalent to the join * [ ]  ∈  CC( ,  ), where is exactly as in Proposition 6.7 and  ⊆ ( ). There is a one-to-one correspondence between the parabolic subgroups occurring in the join and the elements of  ′ ; in particular, One now follows the arguments of Section 6.2.2 to show that if [ ]  is an abelian equivalence class of size , we have CC( ,  ) ≃ CC(GL (ℤ), ) with  ⊆ (GL (ℤ)) and that if [ ]  is a free equivalence class, we have Both CC(GL (ℤ), (GL (ℤ)) ) and CC(Out( ;  ), (Out( ;  )) ) are homotopy Cohen-Macaulay: In the first case, this holds because the coset complex is isomorphic to the building associated to GL (ℚ) (see Proposition 4.3), in the second case this is Theorem 4.21. Hence, Theorem 8.1 implies that CC( ,  ) is spherical of dimension | | − 1. It now follows from Lemma 2.5 that An immediate consequence of this is that CC( , ( )) is a chamber complex, that is, that each pair , of facets of CC( , ( )) can be connected by a sequence of facets = 1 , … , = such that for all 1 ⩽ ⩽ , the intersection of and +1 is a face of codimension 1 (see [4, Proposition 11.7; 9, Remark 2.8]).

8.2
Parabolic subgroups of lower rank Definition 8.3. Let ∶= rk( ) and 1 ⩽ ⩽ − 1. We define the family of rank-standard parabolic subgroups of as the set of all intersections of ( − ) distinct maximal standard parabolic subgroups, In particular, we have ( ) =  −1 ( ).
Every parabolic subgroup of is itself a relative automorphism group of Γ . The term 'rank-' parabolic subgroup is justified by the following proposition. Proof. For every ∈  ( ), there is a ⊂ (Γ) and for every ∈ a subset ⊂ {1, … , |[ ]  |} such that As  contains (), so does  ′ . It is easy to check that if ∈ , the -equivalence class [ ]  can be written as the disjoint union of (| | + 1)-many  ′ -equivalence classes and that otherwise, one has [ ]  = [ ]  ′ . From this, the claim follows immediately. □ Theorem B is now an easy corollary of Cohen-Macaulayness of CC( , ( )) and the results of [9].

Presentations for
A consequence of higher generation is that one can obtain presentations of from presentations of the parabolic subgroups as follows: Write  ( ) = { 1 , … , ( ) }. For each , let be the set of all inversions, transvections and partial conjugations of that are contained in . By Theorem 5.1, the set generates . Let = ⟨ | ⟩ be a presentation for . Then we have the following corollary.
Proof. This follows from Corollary 8.5 and Theorem 3.8. □

Interpretation of rank in terms of Coxeter groups
The rank of a group with a -pair is given by the rank of the associated Weyl group , which is a Coxeter group. This is also true in the setting of relative automorphism groups of RAAGs as we will see in what follows. Definition 8.7. Let Aut(Γ) denote the group of graph automorphisms of Γ. This group embeds in Out( Γ ) and we define Aut 0 (Γ) as the intersection Aut(Γ) ∩ .
If is equal to Out( ) or GL (ℤ), we have Aut 0 (Γ) = Aut(Γ) = Sym( ), the Weyl group associated to GL (ℚ), which has rank − 1. In general, Aut 0 (Γ) can be seen as the group of 'algebraic' graph automorphisms of Γ. It appears as 'Sym 0 (Γ)' in [13,Section 3.2] where it is studied under the additional assumption that Γ be connected and triangle-free. . It remains to show that this group does not contain any other elements, that is, that every element of Aut 0 (Γ) preserves all the -equivalence classes of (Γ). To see this, assume that ∈ Aut( Γ ) represents an element of such that ( ) = ′ for some , ′ ∈ (Γ). We will show that ∼  ′ : For ∈ (Γ) and a word in the alphabet (Γ) ±1 , let # ( ) ∈ ℤ denote the number of occurrences of in , counted with sign. For g ∈ Γ , let # (g) ∈ ℤ∕2ℤ be the image of # ( ) in ℤ∕2ℤ, where is a word representing g -this number only depends on g and not on the chosen representative . Now assume that ≠ ′ . Then clearly, # ′ ( ) = 0 and # ′ ( ( )) = # ′ ( ′ ) = 1. Writing as a product of inversions, transvections and partial conjugations, it follows that there must be such a Laurence generator [ ] ∈ with # ′ ( ( )) = 1. This is only possible if is given by the transvection ′ . However, if this is contained in , we know that ⪯ ′ . As −1 sends ′ to , we also have ′ ⪯ , hence ∼  ′ . □ Recall that the rank of a Coxeter system ( , ) is given by rk( , ) = | |. Corollary 8.9. There is a subset ⊂ Aut 0 (Γ) ⩽ such that (Aut 0 (Γ), ) is a Coxeter system of rank equal to rk( ).
Proof. The symmetric group on a set of elements is the Coxeter group of type −1 , so the claim follows from Lemma 8.8. □ Additional comments on this can be found in the 'BN-pairs' paragraph of Section 9.

CLOSING COMMENTS AND OPEN QUESTIONS
We conclude with comments on the limitations of our constructions and on open questions related to the complex  = CC( , ( )).

Description as a subgroup poset in
Both in the setting of GL (ℤ) and of Fouxe-Rabinovitch groups Out( ;  ), we studied the coset complex of parabolic subgroups by finding an isomorphic poset of subgroups of Γ and then determined its homotopy type. These were the poset of direct summands of ℤ and the relative free factor complex  ( , ), respectively. In general, however, the author is not aware of a natural description of  which looks similar. It is not hard to see that if = Stab ( Δ ) and ′ = Stab ( Δ ′ ) are distinct parabolic subgroups, then the -orbits of [ Δ ] and [ Δ ′ ] intersect trivially. Hence, the map g Stab ( Δ ) ↦ [g( Δ )] defines a bijection between the vertices of  and the union of the -orbits of conjugacy classes of special subgroups Δ that we used for the definition of parabolic subgroups. However, it is not true that the adjacency relation in  is given by containment of corresponding subgroups of Γ . These orbits can be described more explicitly: Let Δ ⊂ Γ be the full subgraph of Γ with vertex set { 1 , … , } ∪ Γ ≻ . As ⪰ is stabilised by , it is clear that } .
If on the other hand [ ] is a free equivalence class, one has Γ ⪰ = Δ * , where ∶= lk( ) ∩ Γ ⪰ is a complete graph and Δ = [ ] ⊔ Γ 1 ⊔ ⋯ ⊔ Γ with Γ 1 , … , Γ the connected components of Γ ≻ ⧵ . Hence, Γ decomposes as a direct product Δ × and every element of preserves this product structure. It follows that the -orbit of [ Δ ] is equal to
Using [26, Lemma 2.11] (see Section 4.2.2), every element in this orbit is of the form * 1 Γ 1 * ⋯ * Γ , where ∈ Δ and is a free group of rank .

Limitations of our construction
It seems that our definition of parabolic subgroups and the corresponding coset complex capture well the aspects of Out( Γ ) that come from similarities of this group with GL (ℤ) and Out( ): Firstly, our definitions recover the Tits building as CC(GL (ℤ), (GL (ℤ))) and the free factor complex as CC(Out( ), (Out( ))). Secondly, the results we obtain show strong similarities in behaviour between the general situation of Out( Γ ) and these special cases: The associated coset complex is spherical, even Cohen-Macaulay (Theorem 8.2) and families of parabolic subgroups are highly generating with the degree of generation depending on the rank of these subgroups (Corollary 8.5). Another strong indication which suggests a certain optimality of our definitions is the description of rk(Out 0 ( Γ )) in terms of a Coxeter subgroup (Corollary 8.9). Furthermore, our induction leads to well-suited families of parabolic subgroups in all those 'components' of Out 0 ( Γ ) that closely resemble general linear groups and automorphism groups of free groups; that is, the base cases that are given by GL (ℤ), ⩾ 2, and Fouxe-Rabinovitch groups containing transvections (Items 2(b) and 3(c) in Section 7.1). However, our construction is rather transvection-based in the sense that the standard ordering of (Γ) -which is used to define the parabolic subgroups -is entirely determined by the transvections that Out( Γ ) contains. This makes our definition of parabolic subgroups quite local: Whether or not ⩽ can be read off from the one-balls around these vertices. This is also reflected by the fact that the conical subgraphs Γ ⩾ , which play a central role in our induction, are contained in the two-balls around if is not an isolated vertex. In contrast, certain aspects of Out( Γ ) seem not to be mere generalisations of phenomena in arithmetic groups and automorphism groups of free groups. For example, Out( Γ ) contains partial conjugations which cannot be written as a product of transvections. The existence of these partial conjugations is a global phenomenon in the sense that the shape of the connected components of Γ ⧵ st( ) is not determined by local conditions on Γ. These aspects are not very well-represented in : The base cases of our induction that correspond to them do not contain any parabolic subgroups. In the extremal case where there is no equivalence class of (Γ) that has size greater than one, (Out 0 ( Γ )) is even empty. For specific applications, one might try to overcome this by introducing further parabolic subgroups that capture these global aspects. However, the author currently does not see a canonical way to do this.

BN-pairs
The existence of a 'Weyl group' Aut 0 (Γ) as described in Section 8.3 suggests that one might be able to transfer additional notions from the theory of groups with BN-pair to automorphism groups of RAAGs. It does, for instance, seem reasonable to define a 'Borel-subgroup' by taking the intersection of all standard parabolic subgroups or to use the Weyl group to define apartments in . For this, it might be helpful to use the standard representation Out( Γ ) → GL | (Γ)| (ℤ) induced by the abelianisation. The question that has yet to be clarified is to what extent this point of view might be fruitful for studying automorphism groups of RAAGs; one should keep in mind that all this can also be done for = Out( ) which is far away from having a BN-pair.

Boundary structures
Both buildings and free factor complexes can be seen as boundary structures of classifying spaces -in the first case, this is due to Borel-Serre who constructed a bordification of symmetric spaces whose boundary can be described by rational Tits buildings [5]; in the second case, it was shown in [10] that the free factor complex can be seen as a subspace of the simplicial boundary of Culler-Vogtmann Outer space. In the RAAG-setting, one may ask whether a similar statement holds and  can be seen as a boundary structure of the RAAG Outer space defined in [6,15] or a similar space. However, without further changes, this will not work for arbitrary . In particular, if does not contain any transvection, the complex  is trivial, while this need not be the case for the RAAG Outer space and its boundary. This, for example, occurs for RAAGs defined by focused graphs that appear in the work of Bregman and Fullarton [7], if the standard ordering on the graph Γ is trivial. In that case, Out 0 ( Γ ) is a semi-direct product of a free abelian group generated by partial conjugations and the (finite) group of inversions.

Geometric aspects
This text focuses on the topology of . It also seems very reasonable, however, to ask what can be said about the geometry of this complex. Motivated by the work of Masur and Minsky [34], who showed that the curve complex ( ) is hyperbolic, Bestvina and Feighn [3] proved that the free factor complex is hyperbolic as well. This is only one of many results in the study of Out( ) from a geometric point of view, which has become popular in recent years. On the other hand, there is also a rich theory concerning metric aspects of buildings (for an overview, see [2,Section 12]). Combining these two theories should be an interesting topic for further investigations.

A C K N O W L E D G E M E N T S
I would like to thank my supervisor Kai-Uwe Bux for his support and guidance throughout this project. Furthermore, many thanks are due to Dawid Kielak and Ric Wade for numerous helpful remarks and suggestions at different stages of this project. I would also like to thank Herbert Abels, Stephan Holz and Yuri Santos Rego for several interesting conversations about coset complexes and Kai-Uwe Bux, Radhika Gupta, Dawid Kielak and Yuri Santos Rego for helpful comments on a first draft of this text. I am also grateful to the anonymous referee for the careful reading and many good suggestions that helped to improve the quality of this text. The author was supported by the grant BU 1224/2-1 within the Priority Programme 2026 'Geometry at infinity' of the German Science Foundation (DFG).

J O U R N A L I N F O R M AT I O N
The Journal of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not-for-profit Charity registered with the UK Charity Commission. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.