# Homological Stability for automorphism groups of Raags

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Homological Stability for automorphism groups of Raags

GIOVANNI GANDINI NATHALIE WAHL

We show that the homology of the automorphism group of a right-angled Artin group stabilizes under taking products with any right-angled Artin group.

20F65; 20F28

Introduction

It has been conjectured that, for any (ﬁnitely generated) discrete group G, the homology groups Hi(Aut(G∗n); Z) and Hi(Aut(Gn); Z) should be independent of n, for n i, generalizing the classical stability results for GLn(Z) and Aut(Fn) when G = Z. (See the conjectures [9, Conjecture 1.4],[15, Conjecture 5.16], and the classical results in [2, 7, 8, 14, 17].) The stabilization of Hi(Aut(G∗n); Z) for i large has been shown to hold for most groups by the main theorem of [3] and [9, Corollary 1.3].1 The stabilization of Hi(Aut(Gn); Z) in contrast has so far only been known in two extreme cases: when G is abelian and when G has trivial center and does not factorize as a direct product. Indeed, in the ﬁrst case Aut(Gn) is isomorphic to GLn(End(G)), which is known to stabilize (see Proposition 5.2), while in the second case, the group Aut(Gn) is isomorphic to Aut(G) Σn [12], a group that is also known to stabilize [9, Proposition 1.6].2 In the present paper, we verify that the second conjecture holds for G any right-angled Artin group, possibly factorizable, possibly with a non-trivial center. This proves a ﬁrst “mixed case” of the conjecture, which interpolates between the two previously known cases.

A right-angled Artin group (or RAAG) is a group with a ﬁnite set of generators s1, . . . , sn and relations that are commutation relations between the generators, i.e. relations of

1[3] gives stability for Aut(G∗n) with G any group with a ﬁnite free product decomposition (eg. a ﬁnitely generated group) without Z factor, while [9] treats the cases with G arising as fundamental groups of certain 3–manifolds, allowing Z factors in the free product decomposition.

2Slightly more generally, for the second case, one can get stability for Aut(Gn) for G a product of certain such center-free groups using [12].

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Giovanni Gandini and Nathalie Wahl

the form sisj = sjsi for certain i’s and j’s. The extreme examples of RAAGs are the free groups Fn if no commutation relation holds, and the free abelian groups Zn if all commutation relations hold. Given any two RAAGs A and B, their product is again a RAAG. We consider in the present paper the sequence of groups Gn = Aut(A × Bn) associated to A and B, and the sequence of maps

σn : Gn = Aut(A × Bn) −→ Gn+1 = Aut(A × Bn+1)

taking an automorphism f of A × Bn to the automorphism f × B of A × Bn+1 leaving the last B factor ﬁxed. Note that when A is the trivial group, the group Gn = Aut(Bn) is a group as in the second conjecture above.

Our main result is the following:

Theorem A (Stability with constant coefﬁcients) Let A, B be any RAAGs. The map

Hi(Aut(A × Bn); Z) −→ Hi(Aut(A × Bn+1); Z)

induced

by

σn

is

surjective

for

all

i

≤

n−1 2

and

an

isomorphism

for

i

≤

n−2 2 .

If

B

has

no

Z –factors,

then

surjectivity

holds

for

i

≤

n 2

and

injectivity

for

i

≤

n−2 1 .

We prove this stability theorem using the general method developed by Randal-Williams and the second author in [15]. This method provides a more general stability result, namely stability in homology not only with constant coefﬁcients Z as above, but also with both polynomial and abelian coefﬁcients, and we establish our main result also in this level of generality as Theorem 5.1. The following theorems are further special cases of Theorem 5.1:

Stability for Aut(A × Bn) with the (abelian) coefﬁcients H1(Aut(A × Bn)) implies the following:

Theorem B (Stability for commutator subgroups) Let A, B be any RAAGs and let Aut (A × Bn) denote the commutator subgroup of Aut(A × Bn). The map

Hi(Aut (A × Bn); Z) −→ Hi(Aut (A × Bn+1); Z)

induced

by

σn

is

surjective

for

all

i

≤

n−2 3

and

an

isomorphism

for

i

≤

n−3 4 .

If

B

has

no

Z –factors,

then

surjectivity

holds

for

i

≤

n−1 3

and

injectivity

for

i

≤

n−3 3 .

An example of a polynomial coefﬁcient system for the groups Aut(A × Bn) is the sequence of “standard” representations H1(A × Bn), and stability with polynomial coefﬁcients yields the following in that case:

Homological Stability for automorphism groups of Raags

3

Theorem C (Stability with coefﬁcients in the standard representation) Let A, B be any RAAGs. Then the map

Hi(Aut(A × Bn); H1(A × Bn)) −→ Hi(Aut(A × Bn+1); H1(A × Bn+1))

is

surjective

for

all

i

≤

n−2 2

and

an

isomorphism

for

i

≤

n−2 3 .

If

B

has

no

Z –factors,

then surjectivity holds for i ≤ n−2 1 and injectivity for i ≤ n−2 2 .

To prove the above theorems, we show that right-angled Artin groups under direct product ﬁt in the set-up of homogeneous categories developed in [15], and recalled here in Section 1. The main ingredient of stability is the high connectivity of certain semisimplicial sets Wn(A, B) associated to the sequence of groups Aut(A × Bn). We deﬁne and study those semisimplicial sets in Section 2, together with three closely related simplicial complexes In(A, B), SIn(A, B) and Sn(A, B). Sections 1 and 2 are written in the general context of families of groups closed under direct product. In Section 3, we show that right-angled Artin groups admit a “prime decomposition” with respect to direct product, and we give a description of the automorphism group of such a group in terms of this decomposition. Section 4 then uses these results that are speciﬁc to RAAGs together with the complexes deﬁned in Section 2 to prove that the semisimplicial sets Wn(A, B) are highly connected. For the connectivity results, we use join complex methods from [9], as well as an argument of Maazen [14] for the case B = Z. Finally Section 5 states the general stability result, which, given the connectivity result, is a direct application of the main result in [15].

Acknowledgement

The second author was supported by the Danish National Sciences Research Council (DNSRC) and the European Research Council (ERC), as well as by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).

1 Families of groups

We consider here families of groups F which are closed under direct product. We say that F satisﬁes cancellation if for all A, B, C in F , we have that

A × C ∼= B × C =⇒ A ∼= B. Cancellation is not satisﬁed for the family of all ﬁnitely generated groups, see eg. [11, Section 3] or [10] for an example where cancellation with Z fails. Cancellation though

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Giovanni Gandini and Nathalie Wahl

holds for the family of all ﬁnitely generated abelian groups by their classiﬁcation, the family of all ﬁnite groups [10], or for the family of all right angled Artin groups as we will show in Section 3.

Given a family of groups F , we let GF denote its associated groupoid, namely the groupoid with objects the elements of F and morphisms all group isomorphisms. Let e denote the trivial group. When F is closed under direct product, we have that (GF , ×, e) is a symmetric monoidal groupoid.

Recall from [15, Section 1.1], [6, page 219] the category UGF = GF , GF associated to (GF , ×, e): it has the same objects as GF , namely the elements of F , and morphisms from A to B given as pairs (X, f ), where X ∈ F and f : X × A −→ B is an isomorphism, up to the equivalence relation that (X, f ) ∼ (X , f ) if there exists an isomorphism φ : X → X such that f = f ◦ (φ × A).

Recall from [15, Deﬁnition 1.2] that a monoidal category (C, ⊕, 0) is called homogeneous if 0 is initial in C and for every A, B in C , the following two properties hold:

H1 Hom(A, B) is a transitive Aut(B)-set under post-composition;

H2 The map Aut(A) → Aut(A ⊕ B) taking f to f ⊕ B is injective with image Fix(B, A ⊕ B),

where Fix(B, A ⊕ B) is the set of φ ∈ Aut(A ⊕ B) satisfying that φ ◦ (ιA ⊕ B) = ιA ⊕ B in Hom(B, A ⊕ B), for ιA : 0 → A the unique morphism.

Proposition 1.1 If F satisﬁes cancellation, then the category UGF is a symmetric monoidal homogeneous category whose underlying groupoid is GF .

Proof As (GF , ×, e) is symmetric monoidal, UGF is symmetric monoidal by [15, Proposition 1.6], and e is initial in UGF . We have that GF satisﬁes cancellation by assumption, and for any A, B ∈ F , the map AutGF (A) → AutGF (A × B) taking f to f × B is injective. Then [15, Theorem 1.8] implies that UGF is a homogeneous category. Finally, if A × B ∼= e, we must have A = B = e and the unit e has no non-trivial automorphisms. Hence GF satisﬁes the hypothesis of Proposition 1.10 in [15], which gives that GF is the underlying groupoid of UGF .

Remark 1.2 If one wants to consider a family F that does not satisfy cancellation, one can replace GF by a groupoid that does satisfy cancellation (by forgetting that certain objects are isomorphic) and obtain an associated homogeneous category. We will however here for simplicity only consider families satisfying cancellation.

Homological Stability for automorphism groups of Raags

5

We end the section by showing that the homogeneous categories UGF considered here are not pathological in the sense that they satisfy the following standardness property: Let (C, ⊕, 0) be a homogeneous category and (A, X) a pair of objects in C . We say that C is locally standard at (A, X) [15, Deﬁnition 2.5] if

LS1 The morphisms ιA ⊕ X ⊕ ιX and ιA⊕X ⊕ X are distinct in Hom(X, A ⊕ X⊕2); LS2 For all n ≥ 1, the map Hom(X, A ⊕ X⊕n−1) → Hom(X, A ⊕ X⊕n) taking f to

f ⊕ ιX is injective.

Proposition 1.3 For any family F , the category UGF is locally standard at any (A, X).

To prove this proposition, it is easiest to use an alternative description of the morphisms in the category UGF , given by the following:

Lemma 1.4 The association [X, f ] → (f (X), f |A) deﬁnes a one-to-one correspondence between HomUGF (A, B) and the set of pairs (H, g) with H ≤ B and g : A → B an injective homomorphism such that B = H × g(A).

Proof of Lemma 1.4 First note that both f (X) and f |A are independent of the representative of [X, f ], so the association is well-deﬁned.

Suppose that [X, f ] and [Y, g] are morphisms from A to B in UGF satisfying that (f (X), f |A) = (g(Y), g|A). Then g−1|f (X) ◦ f |X : X → Y is an isomorphism and f = g ◦ ((g−1|f (X) ◦ f |X) × A) as both maps agree on their restrictions to X and A. Hence [X, f ] = [Y, g].

We are left to check that any (H, g) is in the image. This follows from the fact that, given such an (H, g), the map H × g : H × A → B is an isomorphism.

Proof of Proposition 1.3 We need to check the two axioms LS1 and LS2. For LS1, we need that the maps ιA ×X ×ιX and ιA×X ×X from X to A×X2 in UGF are distinct. From the deﬁnition of the monoidal structure in UGF given in the proof of Proposition 1.6 of [15], we have that ιA × X × ιX = [A × X, A × b−X,1X] and ιA×X × X = [A × X, idA×X2], where bX,X = b−X,1X : X2 → X2 denotes the symmetry. The fact that they are distinct then follows from the lemma as, for example, (A × b−X,1X)|e×e×X = idA×X2|e×e×X . For LS2, we need to show that the map − × ιX : Hom(X, A × Xn−1) → Hom(X, A × Xn) is injective. This follows again from Lemma 1.4 as (H, f ) × ιX = (H × in(X), f ) in the description of the morphisms given by the lemma, where in(X) ≤ A × Xn denotes the last X factor. This association is injective.

6

Giovanni Gandini and Nathalie Wahl

2 Simplicial complexes and semi-simplical sets associated to a family of groups

To a family of groups F closed under direct product, we associated in the previous section a category UGF with objects the elements of F . Using the morphism sets in this category, the paper [15] associates to any pair of objects A, X ∈ F and any n ≥ 0, a semisimplicial set Wn(A, X) and a simplicial complex Sn(A, X). In the present section, we recall the deﬁnitions of Sn(A, X) and Wn(A, X) and introduce new simplicial complexes In(A, X) and SIn(A, X) likewise associated to A, X ∈ F . We then study the relationship between these four different simplicial objects. To prove homological stability, we will need to show that the semisimplicial sets Wn(A, X) are highly connected. This will be done in Section 4 in the case of the family of all right-angled Artin groups using the three simplicial complexes introduced here. We give in the present section results that allow transfer of connectivity from one of the above spaces to another that work in a general context and that will be combined in Section 4 with results speciﬁc to right-angled Artin groups. For simplicity, we will again assume that F satisﬁes cancellation:

Standing assumption for the section: F is a family of ﬁnitely generated groups, closed under direct product, and satisfying cancellation.

Given groups A, X in F , we will consider injective maps f : Xk → A × Xn so that there is a splitting A × Xn = f (Xk) × H with H in F . As F satisﬁes cancellation, we always have that H ∼= A × Xn−k . We call such a map f an F –split map, and we call the pair (f , H) an F –splitting.

Recall that a simplicial complex Y is deﬁned from a set of vertices Y0 by giving a collection of ﬁnite subsets of Y0 closed under taking subsets. The subsets of cardinality p + 1 are called the p–simplices of Y . On the other hand, a semisimplicial set W is a collection of sets Wp of p–simplices for each p ≥ 0 related by boundary maps di : Wp → Wp−1 for each 0 ≤ i ≤ p satisfying the simplicial identities. Both simplicial complexes and semisimplicial sets admit a realization, that has a copy of ∆p for each p–simplex of the simplicial object. When we talk about connectivity of such objects, we always refer to the connectivity of their realization.

We deﬁne now three simplicial complexes and one semisimplicial set whose objects are either F –split maps or F –splittings.

Homological Stability for automorphism groups of Raags

7

Deﬁnition 2.1 To a pair of groups X, A ∈ F and a natural number n ≥ 0, we associate the following simplicial complexes:

In(A, X) A vertex in In(A, X) is an F –split map f : X → A × Xn . Distinct vertices f0, . . . , fp form a p–simplex in In(A, X) if the map (f0, . . . , fp) : Xp+1 → A × Xn is F –split.

SIn(A, X) A vertex in SIn(A, X) is an F –splitting (f , H) with f ∈ In(A, X). Distinct vertices (f0, H0), . . . , (fp, Hp) form a p–simplex of SIn(A, X) if f0, . . . , fp is a p–simplex of In(A, X) and fi(X) ≤ Hj for each i = j.

Sn(A, X) The vertices of Sn(A, X) are the same as those of SIn(A, X). Distinct vertices (f0, H0), . . . , (fp, Hp) form a p–simplex of Sn(A, X) if there exists an F –splitting (f , H), with f = (f0, . . . , fp) : Xp+1 → A × Xn , such that Hj = H × i=j fi(X) for each j.

We moreover associate the following semisimplicial set:

Wn(A, X) A p–simplex in Wn(A, X) is an F –splitting (f , H), with f : Xp+1 → A × Xn , and the jth face dj(f , H) = (f ◦ dj, H × f (ij)) for dj : Xp → Xp+1 the map skipping the (j + 1)st factor and ij = ιXj × X × ιXp−j : X → Xp+1 .

Using Lemma 1.4, one checks immediately that Wn(A, X) identiﬁes with the semisimplicial set of [15, Deﬁnition 2.1] associated to the category UGF , and Sn(A, X) identiﬁes with the simplicial complex of [15, Deﬁnition 2.8] likewise associated to UGF .

The following proposition shows that, in the context we work with, we can always approach the connectivity of Wn(A, X) via that of Sn(A, X).

Proposition 2.2 Let F be a family of groups satisfying cancellation and let a, k ≥ 1. The simplicial complex Sn(A, X) is ( n−k a )–connected for all n ≥ 0 if and only if the semisimplicial set Wn(A, X) is ( n−k a )–connected for all n ≥ 0.

Proof As UGF is symmetric monoidal, homogeneous (Proposition 1.1) and locally standard (Proposition 1.3), Proposition 2.9 of [15] yields that the semisimplicial sets Wn(A, X) satisfy condition (A) in that paper (see [15, Section 2.1]). The result then follows from [15, Theorem 2.10].

Note that there is an inclusion of simplicial complexes Sn(A, X) → SIn(A, X).

8

Giovanni Gandini and Nathalie Wahl

Indeed, the two complexes have the same set of vertices, and simplices of Sn(A, X) satisfy the condition for being a simplex of SIn(A, X). There is also a forgetful map

SIn(A, X) −→ In(A, X).

Recall from [9, Deﬁnition 3.2] that a join complex over a simplicial complex X is a simplicial complex Y together with a simplicial map π : Y → X satisfying the following properties:

(1) π is surjective;

(2) π is injective on individual simplices;

(3) For each p-simplex σ = x0, · · · , xp of X the subcomplex Y(σ) of Y consisting of all the p-simplices that project to σ is the join Yx0(σ) ∗ · · · ∗ Yxp(σ) of the vertex sets Yxi(σ) = Y(σ) ∩ π−1(xi).

We say that Y is a complete join over X if Yxi(σ) = π−1(xi) for each σ and each xi .

Join complexes usually arise via labeling systems (see [9, Example 3.3]): a labeling system for a simplicial complex X is a collection of nonempty sets Lx(σ) for each simplex σ of X and each vertex x of σ , satisfying Lx(τ ) ⊃ Lx(σ) whenever x ∈ τ ⊂ σ . One can think of Lx(σ) as the set of labels of x that are compatible with σ . We can use the labeling system L to deﬁne a new simplicial complex XL having vertices the pairs (x, l) with x ∈ X and l ∈ Lx( x ). A collection of pairs ((x0, l0), · · · , (xp, lp)) then forms a p-simplex of XL if and only if σ = x0, · · · , xp is a p-simplex of X and li ∈ Lxi(σ) for each i. Then the natural map π : XL → X forgetting the labels represents XL as a join complex over X .

Proposition 2.3 The complex SIn(A, X) is a join complex over In(A, X).

Proof We check that SIn(A, X) can be constructed from In(A, X) via a labeling system in the sense described above. For each simplex σ = f0, . . . , fp of In(A, X) and each vertex fi in σ , we deﬁne the set of labels of fi compatible with σ as

Lfi(σ) := {H ≤ A × Xn | (fi, H) ∈ SIn(A, X), fj(X) ≤ H for each fj = fi ∈ σ}.

These sets are non-empty because the fact that f0, . . . , fp is a simplex of In(A, X) implies that there exists an F –splitting (f , H) with

f = (f0, . . . , fp) : Xp+1 → A × Xn = H × f (Xp+1).

Let Hi = H × j=i fj(X). Then Hi ∈ Lfi(σ). We clearly have that for any fi ∈ τ ⊂ σ , Lfi(τ ) ⊃ Lfi(σ), and SIn(A, X) = (In(A, X))L .

Homological Stability for automorphism groups of Raags

9

This will allow us to use results from [9] to obtain in good cases a connectivity bound for SIn(A, X) from one for In(A, X).

We now show that, under one additional assumption, Sn(A, X) and SIn(A, X) are isomorphic, in which case we will also get a connectivity result for Sn from that of SIn .

Proposition 2.4 Suppose that for any simplex (f0, H0), . . . , (fp, Hp) of SIn(A, X), we

have that

p i=0

Hi

∈

F.

Then

the

inclusion

Sn(A, X)

→

SIn(A, X)

is

an

isomorphism.

Lemma 2.5 Suppose A, B, A , B are groups such that A × B = A × B and A ≤ A. Then A = A × (B ∩ A).

Proof Consider the inclusion A × (B ∩ A) → A. This is an injective group homomorphism. Now every a ∈ A ≤ A × B = A × B can be written as a = a b with a ∈ A and b ∈ B . But then b = (a )−1a ∈ A and hence a ∈ A × (B ∩ A) and the map is also surjective.

Proof of Proposition 2.4 Recall that Sn(A, X) and SIn(A, X) have the same set of vertices, and that there is an inclusion Sn(A, X) → SIn(A, X), that is simplices of Sn(A, X) are also simplices in SIn(A, X). So we are left to check that simplices of SIn(A, X) are also always simplices in Sn(A, X). So consider a p–simplex (g0, K0), . . . , (gp, Kp) of SIn(A, X). We have that g = (g0, . . . , gp) : Xp+1 → A × Xn is split injective. To show that these vertices form a p–simplex in Sn(A, X), we need to ﬁnd a complement K ≤ A × Xn for g with K ∈ F satisfying that

(1)

Kj = K × gi(X).

i=j

Note that if K satisﬁes (1), it necessarily is a complement for g as Kj × gj(X) = A × Xn for each j. Let K = j Kj ≤ A × Xn . By the assumption, we have that K ∈ F .

We will now check that it satisﬁes (1), which will ﬁnish the proof. By renaming

the factors, it is enough to prove that (1) holds for j = 0. We do it by induction:

we start with K0 = K0 . Suppose r ≥ 2 and assume that we have proved that

K0 =

r−1 j=0

Kj

×

g1(X)

×

···

×

gr−1 (X ) .

We

have

that

r−1

Kj × g1(X) × · · · × gr−1(X) × g0(X) = K0 × g0(X) = A × Xn = Kr × gr(X).

j=0

Now gr(X) ≤ Kj for all j = 0, . . . , r − 1. Applying the lemma we thus get that

r−1 j=0

Kj

=

gr(X)

×

(

r−1 j=0

Kj)

Kr , which gives the induction step.

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Giovanni Gandini and Nathalie Wahl

The following proposition will also be useful in the sequel:

Proposition 2.6 The action of Aut(A × Xn) on A × Xn induces an action on the complexes In(A, X) and Sn(A, X) which is transitive on the set of p-simplices for every p in both cases. Moreover, the composed map Sn(A, X) → SIn(A, X) → In(A, X) is equivariant with respect to these actions.

Proof The action is induced by post-composition by automorphisms on the split maps f : Xp+1 → A × Xn , and by evaluation on splittings H ≤ A × Xn . The map Sn(A, X) → In(A, X) forgets the choice of splitting and is hence equivariant. For Sn(A, X), transitivity of the action is axiom H1 in the homogeneous category UGF [15, Deﬁnition 1.2], which is satisﬁed by Proposition 1.1. For In(A, X), it follows from the corresponding fact for Sn(A, X) and the fact that every simplex of In(A, X) admits a lift in Sn(A, X).

3 RAAGs and their groups of automorphisms

Now we consider the family F of all right-angled Artin groups, and give in this section a few properties that are particular to these groups and that will allow us to prove the connectivity result necessary for stability. In particular, we show that the family of RAAGs satisﬁes cancellation and give a description of the automorphism group of a direct product of RAAGs in terms of the automorphism groups of its factors. We start by recalling what a RAAG is.

Given a ﬁnite simplicial graph Γ one can associate a group AΓ with one generator v for each vertex of Γ and a commuting relation vw = wv for each edge (v, w) in Γ. Such a group AΓ is called a graph group or more commonly right-angled Artin group. The main theorem of [4] says that the graph describing such a group is unique in the sense that two such groups AΓ and AΓ are isomorphic if and only if the graphs Γ and Γ are isomorphic.

The next proposition says that RAAGs admit a “prime decomposition” with respect to direct product.

Proposition 3.1 Any RAAG AΓ admits a maximal decomposition as AΓ = AΓ1 × · · · × AΓk

with each AΓi a RAAG, and this decomposition is unique up to isomorphism and permutation of the factors.

GIOVANNI GANDINI NATHALIE WAHL

We show that the homology of the automorphism group of a right-angled Artin group stabilizes under taking products with any right-angled Artin group.

20F65; 20F28

Introduction

It has been conjectured that, for any (ﬁnitely generated) discrete group G, the homology groups Hi(Aut(G∗n); Z) and Hi(Aut(Gn); Z) should be independent of n, for n i, generalizing the classical stability results for GLn(Z) and Aut(Fn) when G = Z. (See the conjectures [9, Conjecture 1.4],[15, Conjecture 5.16], and the classical results in [2, 7, 8, 14, 17].) The stabilization of Hi(Aut(G∗n); Z) for i large has been shown to hold for most groups by the main theorem of [3] and [9, Corollary 1.3].1 The stabilization of Hi(Aut(Gn); Z) in contrast has so far only been known in two extreme cases: when G is abelian and when G has trivial center and does not factorize as a direct product. Indeed, in the ﬁrst case Aut(Gn) is isomorphic to GLn(End(G)), which is known to stabilize (see Proposition 5.2), while in the second case, the group Aut(Gn) is isomorphic to Aut(G) Σn [12], a group that is also known to stabilize [9, Proposition 1.6].2 In the present paper, we verify that the second conjecture holds for G any right-angled Artin group, possibly factorizable, possibly with a non-trivial center. This proves a ﬁrst “mixed case” of the conjecture, which interpolates between the two previously known cases.

A right-angled Artin group (or RAAG) is a group with a ﬁnite set of generators s1, . . . , sn and relations that are commutation relations between the generators, i.e. relations of

1[3] gives stability for Aut(G∗n) with G any group with a ﬁnite free product decomposition (eg. a ﬁnitely generated group) without Z factor, while [9] treats the cases with G arising as fundamental groups of certain 3–manifolds, allowing Z factors in the free product decomposition.

2Slightly more generally, for the second case, one can get stability for Aut(Gn) for G a product of certain such center-free groups using [12].

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Giovanni Gandini and Nathalie Wahl

the form sisj = sjsi for certain i’s and j’s. The extreme examples of RAAGs are the free groups Fn if no commutation relation holds, and the free abelian groups Zn if all commutation relations hold. Given any two RAAGs A and B, their product is again a RAAG. We consider in the present paper the sequence of groups Gn = Aut(A × Bn) associated to A and B, and the sequence of maps

σn : Gn = Aut(A × Bn) −→ Gn+1 = Aut(A × Bn+1)

taking an automorphism f of A × Bn to the automorphism f × B of A × Bn+1 leaving the last B factor ﬁxed. Note that when A is the trivial group, the group Gn = Aut(Bn) is a group as in the second conjecture above.

Our main result is the following:

Theorem A (Stability with constant coefﬁcients) Let A, B be any RAAGs. The map

Hi(Aut(A × Bn); Z) −→ Hi(Aut(A × Bn+1); Z)

induced

by

σn

is

surjective

for

all

i

≤

n−1 2

and

an

isomorphism

for

i

≤

n−2 2 .

If

B

has

no

Z –factors,

then

surjectivity

holds

for

i

≤

n 2

and

injectivity

for

i

≤

n−2 1 .

We prove this stability theorem using the general method developed by Randal-Williams and the second author in [15]. This method provides a more general stability result, namely stability in homology not only with constant coefﬁcients Z as above, but also with both polynomial and abelian coefﬁcients, and we establish our main result also in this level of generality as Theorem 5.1. The following theorems are further special cases of Theorem 5.1:

Stability for Aut(A × Bn) with the (abelian) coefﬁcients H1(Aut(A × Bn)) implies the following:

Theorem B (Stability for commutator subgroups) Let A, B be any RAAGs and let Aut (A × Bn) denote the commutator subgroup of Aut(A × Bn). The map

Hi(Aut (A × Bn); Z) −→ Hi(Aut (A × Bn+1); Z)

induced

by

σn

is

surjective

for

all

i

≤

n−2 3

and

an

isomorphism

for

i

≤

n−3 4 .

If

B

has

no

Z –factors,

then

surjectivity

holds

for

i

≤

n−1 3

and

injectivity

for

i

≤

n−3 3 .

An example of a polynomial coefﬁcient system for the groups Aut(A × Bn) is the sequence of “standard” representations H1(A × Bn), and stability with polynomial coefﬁcients yields the following in that case:

Homological Stability for automorphism groups of Raags

3

Theorem C (Stability with coefﬁcients in the standard representation) Let A, B be any RAAGs. Then the map

Hi(Aut(A × Bn); H1(A × Bn)) −→ Hi(Aut(A × Bn+1); H1(A × Bn+1))

is

surjective

for

all

i

≤

n−2 2

and

an

isomorphism

for

i

≤

n−2 3 .

If

B

has

no

Z –factors,

then surjectivity holds for i ≤ n−2 1 and injectivity for i ≤ n−2 2 .

To prove the above theorems, we show that right-angled Artin groups under direct product ﬁt in the set-up of homogeneous categories developed in [15], and recalled here in Section 1. The main ingredient of stability is the high connectivity of certain semisimplicial sets Wn(A, B) associated to the sequence of groups Aut(A × Bn). We deﬁne and study those semisimplicial sets in Section 2, together with three closely related simplicial complexes In(A, B), SIn(A, B) and Sn(A, B). Sections 1 and 2 are written in the general context of families of groups closed under direct product. In Section 3, we show that right-angled Artin groups admit a “prime decomposition” with respect to direct product, and we give a description of the automorphism group of such a group in terms of this decomposition. Section 4 then uses these results that are speciﬁc to RAAGs together with the complexes deﬁned in Section 2 to prove that the semisimplicial sets Wn(A, B) are highly connected. For the connectivity results, we use join complex methods from [9], as well as an argument of Maazen [14] for the case B = Z. Finally Section 5 states the general stability result, which, given the connectivity result, is a direct application of the main result in [15].

Acknowledgement

The second author was supported by the Danish National Sciences Research Council (DNSRC) and the European Research Council (ERC), as well as by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).

1 Families of groups

We consider here families of groups F which are closed under direct product. We say that F satisﬁes cancellation if for all A, B, C in F , we have that

A × C ∼= B × C =⇒ A ∼= B. Cancellation is not satisﬁed for the family of all ﬁnitely generated groups, see eg. [11, Section 3] or [10] for an example where cancellation with Z fails. Cancellation though

4

Giovanni Gandini and Nathalie Wahl

holds for the family of all ﬁnitely generated abelian groups by their classiﬁcation, the family of all ﬁnite groups [10], or for the family of all right angled Artin groups as we will show in Section 3.

Given a family of groups F , we let GF denote its associated groupoid, namely the groupoid with objects the elements of F and morphisms all group isomorphisms. Let e denote the trivial group. When F is closed under direct product, we have that (GF , ×, e) is a symmetric monoidal groupoid.

Recall from [15, Section 1.1], [6, page 219] the category UGF = GF , GF associated to (GF , ×, e): it has the same objects as GF , namely the elements of F , and morphisms from A to B given as pairs (X, f ), where X ∈ F and f : X × A −→ B is an isomorphism, up to the equivalence relation that (X, f ) ∼ (X , f ) if there exists an isomorphism φ : X → X such that f = f ◦ (φ × A).

Recall from [15, Deﬁnition 1.2] that a monoidal category (C, ⊕, 0) is called homogeneous if 0 is initial in C and for every A, B in C , the following two properties hold:

H1 Hom(A, B) is a transitive Aut(B)-set under post-composition;

H2 The map Aut(A) → Aut(A ⊕ B) taking f to f ⊕ B is injective with image Fix(B, A ⊕ B),

where Fix(B, A ⊕ B) is the set of φ ∈ Aut(A ⊕ B) satisfying that φ ◦ (ιA ⊕ B) = ιA ⊕ B in Hom(B, A ⊕ B), for ιA : 0 → A the unique morphism.

Proposition 1.1 If F satisﬁes cancellation, then the category UGF is a symmetric monoidal homogeneous category whose underlying groupoid is GF .

Proof As (GF , ×, e) is symmetric monoidal, UGF is symmetric monoidal by [15, Proposition 1.6], and e is initial in UGF . We have that GF satisﬁes cancellation by assumption, and for any A, B ∈ F , the map AutGF (A) → AutGF (A × B) taking f to f × B is injective. Then [15, Theorem 1.8] implies that UGF is a homogeneous category. Finally, if A × B ∼= e, we must have A = B = e and the unit e has no non-trivial automorphisms. Hence GF satisﬁes the hypothesis of Proposition 1.10 in [15], which gives that GF is the underlying groupoid of UGF .

Remark 1.2 If one wants to consider a family F that does not satisfy cancellation, one can replace GF by a groupoid that does satisfy cancellation (by forgetting that certain objects are isomorphic) and obtain an associated homogeneous category. We will however here for simplicity only consider families satisfying cancellation.

Homological Stability for automorphism groups of Raags

5

We end the section by showing that the homogeneous categories UGF considered here are not pathological in the sense that they satisfy the following standardness property: Let (C, ⊕, 0) be a homogeneous category and (A, X) a pair of objects in C . We say that C is locally standard at (A, X) [15, Deﬁnition 2.5] if

LS1 The morphisms ιA ⊕ X ⊕ ιX and ιA⊕X ⊕ X are distinct in Hom(X, A ⊕ X⊕2); LS2 For all n ≥ 1, the map Hom(X, A ⊕ X⊕n−1) → Hom(X, A ⊕ X⊕n) taking f to

f ⊕ ιX is injective.

Proposition 1.3 For any family F , the category UGF is locally standard at any (A, X).

To prove this proposition, it is easiest to use an alternative description of the morphisms in the category UGF , given by the following:

Lemma 1.4 The association [X, f ] → (f (X), f |A) deﬁnes a one-to-one correspondence between HomUGF (A, B) and the set of pairs (H, g) with H ≤ B and g : A → B an injective homomorphism such that B = H × g(A).

Proof of Lemma 1.4 First note that both f (X) and f |A are independent of the representative of [X, f ], so the association is well-deﬁned.

Suppose that [X, f ] and [Y, g] are morphisms from A to B in UGF satisfying that (f (X), f |A) = (g(Y), g|A). Then g−1|f (X) ◦ f |X : X → Y is an isomorphism and f = g ◦ ((g−1|f (X) ◦ f |X) × A) as both maps agree on their restrictions to X and A. Hence [X, f ] = [Y, g].

We are left to check that any (H, g) is in the image. This follows from the fact that, given such an (H, g), the map H × g : H × A → B is an isomorphism.

Proof of Proposition 1.3 We need to check the two axioms LS1 and LS2. For LS1, we need that the maps ιA ×X ×ιX and ιA×X ×X from X to A×X2 in UGF are distinct. From the deﬁnition of the monoidal structure in UGF given in the proof of Proposition 1.6 of [15], we have that ιA × X × ιX = [A × X, A × b−X,1X] and ιA×X × X = [A × X, idA×X2], where bX,X = b−X,1X : X2 → X2 denotes the symmetry. The fact that they are distinct then follows from the lemma as, for example, (A × b−X,1X)|e×e×X = idA×X2|e×e×X . For LS2, we need to show that the map − × ιX : Hom(X, A × Xn−1) → Hom(X, A × Xn) is injective. This follows again from Lemma 1.4 as (H, f ) × ιX = (H × in(X), f ) in the description of the morphisms given by the lemma, where in(X) ≤ A × Xn denotes the last X factor. This association is injective.

6

Giovanni Gandini and Nathalie Wahl

2 Simplicial complexes and semi-simplical sets associated to a family of groups

To a family of groups F closed under direct product, we associated in the previous section a category UGF with objects the elements of F . Using the morphism sets in this category, the paper [15] associates to any pair of objects A, X ∈ F and any n ≥ 0, a semisimplicial set Wn(A, X) and a simplicial complex Sn(A, X). In the present section, we recall the deﬁnitions of Sn(A, X) and Wn(A, X) and introduce new simplicial complexes In(A, X) and SIn(A, X) likewise associated to A, X ∈ F . We then study the relationship between these four different simplicial objects. To prove homological stability, we will need to show that the semisimplicial sets Wn(A, X) are highly connected. This will be done in Section 4 in the case of the family of all right-angled Artin groups using the three simplicial complexes introduced here. We give in the present section results that allow transfer of connectivity from one of the above spaces to another that work in a general context and that will be combined in Section 4 with results speciﬁc to right-angled Artin groups. For simplicity, we will again assume that F satisﬁes cancellation:

Standing assumption for the section: F is a family of ﬁnitely generated groups, closed under direct product, and satisfying cancellation.

Given groups A, X in F , we will consider injective maps f : Xk → A × Xn so that there is a splitting A × Xn = f (Xk) × H with H in F . As F satisﬁes cancellation, we always have that H ∼= A × Xn−k . We call such a map f an F –split map, and we call the pair (f , H) an F –splitting.

Recall that a simplicial complex Y is deﬁned from a set of vertices Y0 by giving a collection of ﬁnite subsets of Y0 closed under taking subsets. The subsets of cardinality p + 1 are called the p–simplices of Y . On the other hand, a semisimplicial set W is a collection of sets Wp of p–simplices for each p ≥ 0 related by boundary maps di : Wp → Wp−1 for each 0 ≤ i ≤ p satisfying the simplicial identities. Both simplicial complexes and semisimplicial sets admit a realization, that has a copy of ∆p for each p–simplex of the simplicial object. When we talk about connectivity of such objects, we always refer to the connectivity of their realization.

We deﬁne now three simplicial complexes and one semisimplicial set whose objects are either F –split maps or F –splittings.

Homological Stability for automorphism groups of Raags

7

Deﬁnition 2.1 To a pair of groups X, A ∈ F and a natural number n ≥ 0, we associate the following simplicial complexes:

In(A, X) A vertex in In(A, X) is an F –split map f : X → A × Xn . Distinct vertices f0, . . . , fp form a p–simplex in In(A, X) if the map (f0, . . . , fp) : Xp+1 → A × Xn is F –split.

SIn(A, X) A vertex in SIn(A, X) is an F –splitting (f , H) with f ∈ In(A, X). Distinct vertices (f0, H0), . . . , (fp, Hp) form a p–simplex of SIn(A, X) if f0, . . . , fp is a p–simplex of In(A, X) and fi(X) ≤ Hj for each i = j.

Sn(A, X) The vertices of Sn(A, X) are the same as those of SIn(A, X). Distinct vertices (f0, H0), . . . , (fp, Hp) form a p–simplex of Sn(A, X) if there exists an F –splitting (f , H), with f = (f0, . . . , fp) : Xp+1 → A × Xn , such that Hj = H × i=j fi(X) for each j.

We moreover associate the following semisimplicial set:

Wn(A, X) A p–simplex in Wn(A, X) is an F –splitting (f , H), with f : Xp+1 → A × Xn , and the jth face dj(f , H) = (f ◦ dj, H × f (ij)) for dj : Xp → Xp+1 the map skipping the (j + 1)st factor and ij = ιXj × X × ιXp−j : X → Xp+1 .

Using Lemma 1.4, one checks immediately that Wn(A, X) identiﬁes with the semisimplicial set of [15, Deﬁnition 2.1] associated to the category UGF , and Sn(A, X) identiﬁes with the simplicial complex of [15, Deﬁnition 2.8] likewise associated to UGF .

The following proposition shows that, in the context we work with, we can always approach the connectivity of Wn(A, X) via that of Sn(A, X).

Proposition 2.2 Let F be a family of groups satisfying cancellation and let a, k ≥ 1. The simplicial complex Sn(A, X) is ( n−k a )–connected for all n ≥ 0 if and only if the semisimplicial set Wn(A, X) is ( n−k a )–connected for all n ≥ 0.

Proof As UGF is symmetric monoidal, homogeneous (Proposition 1.1) and locally standard (Proposition 1.3), Proposition 2.9 of [15] yields that the semisimplicial sets Wn(A, X) satisfy condition (A) in that paper (see [15, Section 2.1]). The result then follows from [15, Theorem 2.10].

Note that there is an inclusion of simplicial complexes Sn(A, X) → SIn(A, X).

8

Giovanni Gandini and Nathalie Wahl

Indeed, the two complexes have the same set of vertices, and simplices of Sn(A, X) satisfy the condition for being a simplex of SIn(A, X). There is also a forgetful map

SIn(A, X) −→ In(A, X).

Recall from [9, Deﬁnition 3.2] that a join complex over a simplicial complex X is a simplicial complex Y together with a simplicial map π : Y → X satisfying the following properties:

(1) π is surjective;

(2) π is injective on individual simplices;

(3) For each p-simplex σ = x0, · · · , xp of X the subcomplex Y(σ) of Y consisting of all the p-simplices that project to σ is the join Yx0(σ) ∗ · · · ∗ Yxp(σ) of the vertex sets Yxi(σ) = Y(σ) ∩ π−1(xi).

We say that Y is a complete join over X if Yxi(σ) = π−1(xi) for each σ and each xi .

Join complexes usually arise via labeling systems (see [9, Example 3.3]): a labeling system for a simplicial complex X is a collection of nonempty sets Lx(σ) for each simplex σ of X and each vertex x of σ , satisfying Lx(τ ) ⊃ Lx(σ) whenever x ∈ τ ⊂ σ . One can think of Lx(σ) as the set of labels of x that are compatible with σ . We can use the labeling system L to deﬁne a new simplicial complex XL having vertices the pairs (x, l) with x ∈ X and l ∈ Lx( x ). A collection of pairs ((x0, l0), · · · , (xp, lp)) then forms a p-simplex of XL if and only if σ = x0, · · · , xp is a p-simplex of X and li ∈ Lxi(σ) for each i. Then the natural map π : XL → X forgetting the labels represents XL as a join complex over X .

Proposition 2.3 The complex SIn(A, X) is a join complex over In(A, X).

Proof We check that SIn(A, X) can be constructed from In(A, X) via a labeling system in the sense described above. For each simplex σ = f0, . . . , fp of In(A, X) and each vertex fi in σ , we deﬁne the set of labels of fi compatible with σ as

Lfi(σ) := {H ≤ A × Xn | (fi, H) ∈ SIn(A, X), fj(X) ≤ H for each fj = fi ∈ σ}.

These sets are non-empty because the fact that f0, . . . , fp is a simplex of In(A, X) implies that there exists an F –splitting (f , H) with

f = (f0, . . . , fp) : Xp+1 → A × Xn = H × f (Xp+1).

Let Hi = H × j=i fj(X). Then Hi ∈ Lfi(σ). We clearly have that for any fi ∈ τ ⊂ σ , Lfi(τ ) ⊃ Lfi(σ), and SIn(A, X) = (In(A, X))L .

Homological Stability for automorphism groups of Raags

9

This will allow us to use results from [9] to obtain in good cases a connectivity bound for SIn(A, X) from one for In(A, X).

We now show that, under one additional assumption, Sn(A, X) and SIn(A, X) are isomorphic, in which case we will also get a connectivity result for Sn from that of SIn .

Proposition 2.4 Suppose that for any simplex (f0, H0), . . . , (fp, Hp) of SIn(A, X), we

have that

p i=0

Hi

∈

F.

Then

the

inclusion

Sn(A, X)

→

SIn(A, X)

is

an

isomorphism.

Lemma 2.5 Suppose A, B, A , B are groups such that A × B = A × B and A ≤ A. Then A = A × (B ∩ A).

Proof Consider the inclusion A × (B ∩ A) → A. This is an injective group homomorphism. Now every a ∈ A ≤ A × B = A × B can be written as a = a b with a ∈ A and b ∈ B . But then b = (a )−1a ∈ A and hence a ∈ A × (B ∩ A) and the map is also surjective.

Proof of Proposition 2.4 Recall that Sn(A, X) and SIn(A, X) have the same set of vertices, and that there is an inclusion Sn(A, X) → SIn(A, X), that is simplices of Sn(A, X) are also simplices in SIn(A, X). So we are left to check that simplices of SIn(A, X) are also always simplices in Sn(A, X). So consider a p–simplex (g0, K0), . . . , (gp, Kp) of SIn(A, X). We have that g = (g0, . . . , gp) : Xp+1 → A × Xn is split injective. To show that these vertices form a p–simplex in Sn(A, X), we need to ﬁnd a complement K ≤ A × Xn for g with K ∈ F satisfying that

(1)

Kj = K × gi(X).

i=j

Note that if K satisﬁes (1), it necessarily is a complement for g as Kj × gj(X) = A × Xn for each j. Let K = j Kj ≤ A × Xn . By the assumption, we have that K ∈ F .

We will now check that it satisﬁes (1), which will ﬁnish the proof. By renaming

the factors, it is enough to prove that (1) holds for j = 0. We do it by induction:

we start with K0 = K0 . Suppose r ≥ 2 and assume that we have proved that

K0 =

r−1 j=0

Kj

×

g1(X)

×

···

×

gr−1 (X ) .

We

have

that

r−1

Kj × g1(X) × · · · × gr−1(X) × g0(X) = K0 × g0(X) = A × Xn = Kr × gr(X).

j=0

Now gr(X) ≤ Kj for all j = 0, . . . , r − 1. Applying the lemma we thus get that

r−1 j=0

Kj

=

gr(X)

×

(

r−1 j=0

Kj)

Kr , which gives the induction step.

10

Giovanni Gandini and Nathalie Wahl

The following proposition will also be useful in the sequel:

Proposition 2.6 The action of Aut(A × Xn) on A × Xn induces an action on the complexes In(A, X) and Sn(A, X) which is transitive on the set of p-simplices for every p in both cases. Moreover, the composed map Sn(A, X) → SIn(A, X) → In(A, X) is equivariant with respect to these actions.

Proof The action is induced by post-composition by automorphisms on the split maps f : Xp+1 → A × Xn , and by evaluation on splittings H ≤ A × Xn . The map Sn(A, X) → In(A, X) forgets the choice of splitting and is hence equivariant. For Sn(A, X), transitivity of the action is axiom H1 in the homogeneous category UGF [15, Deﬁnition 1.2], which is satisﬁed by Proposition 1.1. For In(A, X), it follows from the corresponding fact for Sn(A, X) and the fact that every simplex of In(A, X) admits a lift in Sn(A, X).

3 RAAGs and their groups of automorphisms

Now we consider the family F of all right-angled Artin groups, and give in this section a few properties that are particular to these groups and that will allow us to prove the connectivity result necessary for stability. In particular, we show that the family of RAAGs satisﬁes cancellation and give a description of the automorphism group of a direct product of RAAGs in terms of the automorphism groups of its factors. We start by recalling what a RAAG is.

Given a ﬁnite simplicial graph Γ one can associate a group AΓ with one generator v for each vertex of Γ and a commuting relation vw = wv for each edge (v, w) in Γ. Such a group AΓ is called a graph group or more commonly right-angled Artin group. The main theorem of [4] says that the graph describing such a group is unique in the sense that two such groups AΓ and AΓ are isomorphic if and only if the graphs Γ and Γ are isomorphic.

The next proposition says that RAAGs admit a “prime decomposition” with respect to direct product.

Proposition 3.1 Any RAAG AΓ admits a maximal decomposition as AΓ = AΓ1 × · · · × AΓk

with each AΓi a RAAG, and this decomposition is unique up to isomorphism and permutation of the factors.

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