# Automorphisms of Right-Angled Coxeter Groups

Authors: Mauricio Gutierrez; Anton Kaul
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

If(W,S) is a right-angled Coxeter system, then Aut(W) is a semidirect product of the group Aut∘(W) of symmetric automorphisms by the automorphism group of a certain groupoid. We show that, under mild conditions, Aut∘(W) is a semidirect product of Inn(W) by the quotient Out∘(W)=Aut∘(W)/Inn(W). We also give sufficient conditions for the compatibility of the two semidirect products. When this occurs there is an induced splitting of the sequence 1→Inn(W)→Aut(W)→Out(W)→1 and consequently, all group extensions 1→W→G→Q→1 are trivial.


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## Body

## 1. Introduction

A  Coxeter groupW is determined by its diagram Γ. It is known that in certain cases,  W determines Γ as well (see, e.g., [1, 2]). This is the case for
right-angled Coxeter groups [3, 4], where the only relations are s2=1 for all generators s and st=ts for some pairs of generators s and t. For right-angled Coxeter groups, it is
convenient to consider the Coxeter diagram (rather than the classical Coxeter
graph): the presence of an edge with endpoints s and t means that s and t commute in W.The properties of a right-angled Coxeter groupW depend almost exclusively on the combinatorics
of the diagram Γ. This is especially evident in the study of Aut(W). For example, the groupoid 𝔉(Γ) consisting of the vertex sets of complete
subgraphs of Γ plays an important role in [5] where Tits exhibits a split
exact sequence(1.1)1→Aut∘(W)→Aut(W)→Aut(𝔉(Γ))→1. Tits also
defines W to have propriété I if the complementary graph ΓC has no triangles. He goes on to show that if W has propriété I, then Aut∘(W) is isomorphic to Inn(W).In the present article, we focus on the set𝔐(Γ) whose members are the vertex sets of maximal complete subgraphs. We say that Γ has condition C if there exist ℤ∈𝔐(Γ) and a collection T1,…,Tn∈𝔐(Γ)−{Z} such that
(C1)
Ti∩Z≠∅ for 1≤i≤n,(C2)
for eachv∈Z, the cardinality of the set {i∣v∉Ti} is odd.
When Γ has condition C, the subgraph spanned by Z∪T1∪⋯∪Tn has propriété I. Thus, our condition C is a
“local version” of Tits' propriété I.Motivated by the results of Tits regarding sequence
(1.1), we consider the sequence(1.2)1→Inn(W)→Aut(W)→Out(W)→1.A splitting of this sequence
implies that all extensions of the form(1.3)1→W→G→Q→1are trivial (cf. [6]). Clearly (1.2) does not
always split. For example, ℤ2 is a right-angled Coxeter group and(1.4)0→ℤ2→ℤ4→ℤ2→0is a nontrivial extension. On
the other hand, if W has no center, then finding nontrivial
extensions of W is surprisingly difficult. Whether (1.2) splits
for all W with trivial center is currently an open
question.The groupAut∘(W) has been studied extensively in [5, 7] and is called the group of
symmetric automorphisms of W. We approach the problem of whether (1.2) splits by
considering the following.(a)
Does the sequence(1.5)1→Inn(W)→Aut∘(W)→Aut∘(W)/Inn(W)→1split?(b)
Are the splittings of (1.1) and (1.5) compatible?
A positive
answer to both (a) and (b) implies that (1.2) is a split extension. We show in
Theorem 5.4 that if Γ has condition C, then (1.5) splits. To obtain a
splitting of (1.2), we show that the action of Aut(𝔉(Γ)) on Aut∘(W) is compatible if Γ is not “too symmetrical.” More
precisely, our main result is the following.Theorem 1.1.
IfΓ has condition C and each α∈Aut(𝔉(Γ)) leaves all vertices of Z invariant, then (1.2) is a split exact sequence.It was recently
shown [8] that (1.5)
always splits. However, this result does not lead to a generalization of
Theorem 1.1 as the splitting given there is not, in
general, compatible with (1.1).


## 2. Right-Angled Coxeter Groups

Coxeter groups are typically defined by presentations,
and there are various conventions for representing such presentations
diagramatically. In this section, we review some definitions and important
properties, focusing exclusively on the right-angled case. See [9] or [10] for a comprehensive treatment.IfX is any set, let P2(X) denote the set of subsets of X with cardinality 2.
Definition 2.1.
Given a finite setS and E⊆P2(S), let Γ=(S,E) denote the undirected graph with vertex set S and edge set E (note that Γ does not have loops or parallel edges). As
such graphs are often used to represent right-angled Coxeter groups, Γ is called a Coxeter diagram.Definition 2.2.
GivenT⊆S, set ET=E∩P2(T). The graph ΓT=(T,ET) is the subgraph of Γspanned by T. A complete subgraph is maximal if it is not properly contained in
any complete subgraph of Γ.Definition 2.3.
The presentation(2.1)𝒫(Γ)=〈S∣s2,(tu)2;s∈S,{t,u}∈E〉is the Coxeter presentation defined byΓ.Definition 2.4.
A groupW is a right-angled Coxeter group if it has a
presentation 𝒫(Γ) defined by some Coxeter diagram Γ. In this case, one
writes  W=W(Γ) and calls W the right-angled Coxeter group defined by Γ. The pair (W,S) is a right-angled Coxeter system.Remark 2.5.
The Coxeter diagram is not the
same as the traditional Dynkin diagram. Indeed, as graphs, the Coxeter and
Dynkin diagrams are complementary.Clearly, each Coxeter diagram defines a unique
right-angled Coxeter group. On the other hand, to recover the diagram from a
group one must first choose a particular Coxeter presentation. It is natural to
wonder whether nonisomorphic diagrams might define isomorphic groups. The relationship
between right-angled Coxeter groups and their diagrams is clarified by the
following result.Theorem 2.6 (Radcliffe [3]).
If(W,S) and (W,S′) are right-angled Coxeter systems for W, then there is an automorphism ρ:W→W such that ρ(S)=S′.A similar result
was also obtained by Castella in [4].
Definition 2.7.
A subgroup ofW generated by a subset of S is called a special subgroup. If T⊆S, it is customary to write WT for the subgroup generated by T. Finite special subgroups are called spherical subgroups. A subgroup of W is parabolic if it is conjugate to a special
subgroup.We conclude this section with statements of some of
the remarkable properties enjoyed by special subgroups. For proofs, see
[9] or [10].
Theorem 2.8.
IfA,B⊂S, then WA∩WB=WA∩B.Theorem 2.9.
IfT⊆S, then WT is the right-angled Coxeter group defined by ΓT.Corollary 2.10.
The
following are equivalent:
(a)
WT is spherical;(b)
WT is an elementary abelian 2-group of rank |T|;(c)
ΓT is complete.


## 3. Automorphisms of Right-angled Coxeter Groups

For the remainder of this article, letW=W(Γ) be the right-angled Coxeter group defined by
the connected Coxeter diagram Γ=(S,E). It is easily verified that, under the
operation of symmetric difference, the set(3.1)𝔉(Γ)={T⊆S∣WTisfinite}is a commutative groupoid with identity ∅.In [5], Tits uses the group of automorphisms of 𝔉(Γ) to exhibit Aut(W) as a semidirect product. We sketch the
construction. Let σ∈Aut(W). It is well known that every maximal finite
subgroup of W is parabolic (see, e.g., [11, Lemma 3.2.1]).
Consequently, every finite subgroup of W is conjugate into a spherical subgroup. It
follows that, for each T∈𝔉(Γ), there is a unique minimal T¯∈𝔉(Γ) such that σ(WT) is conjugate into WT¯. The map(3.2)q:Aut(W)→Aut(𝔉(Γ))given by q(σ)(T)=T¯ is an epimorphism.
Definition 3.1.
LetAut∘(W) be the kernel of q. Elements of Aut∘(W) are called symmetric automorphisms of W.Givenα∈Aut(𝔉(Γ)), consider α^∈Aut(W) defined by(3.3)α^(s)=∏t∈α(s)tfor all s∈S. A main result of [5] is the following
theorem.
Theorem 3.2.
The mappingAut(𝔉(Γ))→Aut(W) given by α↦α^ is a splitting of the sequence(3.4)1→Aut∘(W)→Aut(W)→Aut(𝔉(Γ))→1.LetdΓ be the standard path metric on Γ that assigns length one to each edge. For each s∈S, define(3.5)s*={t∈S∣dΓ(s,t)≤1},s⊥={t∈S∣dΓ(s,t)≥2}. The subgraph Γs⊥ spanned by the elements of s⊥ gives rise to certain generators of Aut∘(W) as follows. If K is the vertex set of a connected component of Γs⊥, then the map σ:S→W given by(3.6)σ(t)={sts,t∈K,t,t∉K,extends to a unique involution σsK∈Aut∘(W). The following is easily deduced from [7].
Theorem 3.3.
For eachs∈S, let K1s,…,Kmss be the vertex sets of the components of Γs⊥. Then Aut∘(W) is generated by the set {σsKis∣s∈S,1≤i≤ms}.Remark 3.4.
In [7],
Mühlherr gives a complete presentation for Aut∘(W) based on a slightly different set of
generators.


## 4. Symmetric Automorphisms

The subsets ofS that generate maximal spherical subgroups of W play a key role in the subsequent development.
As such, we define(4.1)𝔐(Γ)={T⊆S∣WTisamaximalfinitesubgroup}.Note that 𝔐(Γ) is in one-to-one correspondence with the
family of maximal complete subgraphs of Γ.
The global behavior of a symmetric automorphism is governed by the following
observation: if ϕ∈Aut∘(W) and T∈𝔐(Γ), then there exists an element aT=aT(ϕ)∈W such that ϕ(x)=aTxaT−1 for all x∈WT.Remark 4.1.
WhenT∈𝔐(Γ),WT is its own centralizer. Thus, the element aT above is determined up to right multiplication
by any member of WT (i.e., up to choice of representative for the
coset aTWT).IfT,U∈𝔐(Γ) and T∩U≠∅, then aTxaT−1=aUxaU−1 for all x∈WT∩WU=WT∩U. Consequently, aT−1aU lies in the centralizer of WT∩U.Definition 4.2.
Withϕ,T,U as above, let γTU=γTU(ϕ)=aT−1aU. A representative of the double coset WTγTUWU is called a ϕ-transition
from WT to WU.Remark 4.3.
IfT,U,V∈𝔐(Γ) have pairwise nonempty intersection, then γTT=1,γVU=(γUV)−1, and γTUγUV=γTV. As the terminology suggests, the transitions
are in some sense a group-theoretic analogue of the change-of-coordinate maps
on a manifold.For the remainder of this article we fix an elementZ∈𝔐(Γ). For much of what follows, Z may be chosen arbitrarily; in Section 5 we
show that a preferred choice may exist.Givenx∈W, let ψx be the inner automorphism of W given by w↦xwx−1 for all w∈W. Restricting our attention to Z we define(4.2)Inn(Z)={ψs∈Inn(W)∣s∈WZ},Fix∘(Z)={ϕ∈Aut∘(W)∣ϕ(x)=x∀x∈Z}. In other words, Fix∘(Z) is the pointwise stabilizer of WZ under the action of Aut∘(W) on W. Observe that, since the subgraph ΓZ spanned by Z is complete, Inn(Z) is Abelian.Clearly,Inn(Z) is a subgroup of Fix∘(Z). Let π be the restriction to Fix∘(Z) of the natural map Aut∘(W)→Out∘(W) and choose a class [ϕ]∈Out∘(W) with representative ϕ∈Aut∘(W). If f is the restriction of ϕ to WZ, then f(x)=aZxaZ−1 for all x∈WZ. Since [f−1ϕ]=[ϕ] and f−1ϕ∈Fix∘(Z), it follows that π is onto. We have established the
following.
Theorem 4.4.
The sequence(4.3)0→Inn(Z)→iFix∘(Z)→πOut∘(W)→1is a central extension.In Section5, we construct a retraction of the mapping Inn(Z)→iFix∘(Z). We now describe the key ingredient used in
this construction. For each T∈𝔉(Γ), let(4.4)C(T)=⋂v∈Tv*.It follows at once that WC(T∩Z) is the centralizer of WT∩Z in W (cf. [5, page 350]). The function(4.5)πT:C(T∩Z)→Z−Tgiven by(4.6)πT(s)={1,s∉Z−T,s,s∈Z−T,extends uniquely to a retraction WC(T∩Z)→WZ−T, which we also denote by πT.


## 5. Splittings

As in Section4, we assume that Z is a fixed element of 𝔐(Γ).Definition 5.1.
One says thatZ satisfies condition C if there exist elements T1,T2,…,Tn∈𝔐(Γ)−{Z} such that the following conditions hold.
(C1)
Ti∩Z≠∅ for each 1≤i≤n.(C2)
For eachv∈Z, the cardinality of the set {i∣v∉Ti} is odd.
If Γ contains a maximal complete subgraph Ω whose vertex set Z satisfies condition C, then one
says that Γ has condition C. When Γ has condition C, then for each 1≤i≤n above let Δi denote the maximal complete subgraph of Γ spanned by Ti.Note that, since Γ is connected, when n=0 our hypotheses imply that Γ is complete. In this case, W is Abelian and the results below are trivial.
Thus, we may assume that n is a positive integer.
Example 5.2.
We illustrate condition C with some examples.
(a)
IfΓ contains a maximal complete subgraph Ω with an even number of vertices each of which
meets exactly one other complete subgraph of Γ, then Γ has condition C. For example, any Coxeter diagram Γ that contains Figure 1(a) as a subgraph (where Ω, Δ1,
and Δ2 are maximal complete subgraphs of Γ) has condition C.(b)
Supposen is odd and Γ has maximal complete subgraphs Ω,Δ1,…,Δn. If the  vertex set of Ω is Z={v1,…,vn} and the vertex set of Δi contains every element of Z but vi, then Γ has condition C. For example, any Coxeter diagram Γ that contains Figure 1(b) as a subgraph (where Ω, Δ1,Δ2, and Δ3 are maximal complete subgraphs of Γ) has condition C.(a)(b)Remark 5.3.
In [5], Tits says that a right-angled Coxeter group with
Coxeter diagram Γ has “propriété I” if the complementary
graph ΓC has no triangles. In this case, the inclusion Inn(W)→Aut∘(W) is an isomorphism. If Γ has condition C, then the union Δ1∪⋯∪Δn∪Ω defines a right-angled Coxeter group that has
propriété I. Thus, condition C is in some sense a “local” version of
Tits' propriété I.AssumeΓ satisfies condition C and let ϕ∈Fix∘(Z). In this case, we can choose aZ(ϕ)=1. Then, for each T∈𝔐(Γ) with T∩Z≠∅, the transition γZT(ϕ)=aT(ϕ) is an element of the coset aTWT. It must be emphasized that no left
multiplication by elements of WZ is permitted.
Theorem 5.4.
IfΓ has condition C, then the sequence(5.1)1→Inn(W)→Aut∘(W)→Out∘(W)→1splits.Proof.
LetZ,T1,…,Tn∈𝔐(Γ) satisfying (C1) and (C2) of Definition 5.1.
For each ϕ∈Fix∘(Z), define an inner automorphism r(ϕ)=ψx where(5.2)x=∏i=1nπTi(γZTi(ϕ))=∏i=1nπTi(aTi(ϕ))(recall that ψx is the inner automorphism that conjugates each
element of W by x). By definition, πTi(γZTi(ϕ))=πTi(aTi(ϕ)) lies in WZ−Ti⊆WZ.
Consequently, the terms in the product (5.2) commute and so the mapping(5.3)r:Fix∘(Z)→Inn(Z)is well defined. To see that r is a homomorphism, choose ϕ,κ∈Fix∘(Z). Then, for each Ti we have(5.4)aTi(ϕκ)=ϕ(aTi(κ))⋅aTi(ϕ).Since aTi(κ) lies in the centralizer of WTi∩Z, any reduced expression for aTi(κ) is of the form aTi(κ)=v1⋯vm, where each vj∈C(Ti∩Z). Observe that(5.5)πTi(ϕ(vj))={vj,ifvj∈Z−Ti,1,ifvj∉Z−Ti,=πTi(vj)and so πTi(ϕ(aTi(κ)))=πTi(aTi(κ)). It follows from (5.4) that(5.6)πTi(aTi(ϕκ))=πTi(aTi(κ))⋅πTi(aTi(ϕ)).Now, using the fact that the
image of each πTi lies in the Abelian subgroup WZ, we have that r(ϕκ)=r(ϕ)r(κ).
To see thatr is a retraction, let v∈Z and let k be a number of Ti's that do not contain v. If ψv is the inner automorphism of W that conjugates by v, then aTi(ψv)=v for every 1≤i≤n. From (5.2) we have that r(ψv) conjugates every element by(5.7)∏i=1nπTi(aTi(ψv))=∏i=1nπTi(v)=vk.Since k is odd, vk=v and so r(ψv)=ψv.
Sincer is a retraction and the sequence (4.3) is
central, the mapping(5.8)Fix∘(Z)→Inn(Z)×ker(r)defined by(5.9)ϕ↦(r(ϕ),r(ϕ)−1ϕ)is an isomorphism. Consequently, j=π|ker(r) is an isomorphism from ker(r) onto Out∘(W). A section Out∘(W)→Aut∘(W) is given by the composition h∘j−1, where h:ker(r)→Aut∘(W) is the inclusion(5.10)ker(r)⊆Fix∘(Z)⊆Aut∘(W).As noted in the introduction, to obtain a splitting of
sequence (1.2), Γ must satisfy an “asymmetry” condition.
This is obtained by imposing a restriction on the action of Aut(𝔉(Γ)), and motivates the following
definition.
Definition 5.5.
Letα∈Aut(𝔉(Γ)) and T∈𝔐(Γ). One says that αfixesT if α({t})={t} for every t∈T.Note that ifα fixesT, thenα(T′)=T′ for everyT′⊆T.Theorem 5.6.
IfΓ has condition C and each α∈Aut(𝔉(Γ)) fixes Z, then(5.11)1→Inn(W)→Aut(W)→Out(W)→1splits.Proof.
LetFix(Z)={β∈Aut(W)∣β(z)=zforallz∈Z}. Since each α∈Aut(𝔉(Γ)) fixes Z, it follows that α^ (as defined in (3.3) above) lies in Fix(Z) and the mapping α↦α^ is a splitting of the sequence(5.12)1→Fix∘(Z)→Fix(Z)→q¯Aut(𝔉(Γ))→1(where q¯ is the restriction of the mapping q given in (3.2) above). As in the proof of Theorem 4.4, the projection Fix(Z)→Out(W) is onto and its
kernel is(5.13)Inn(Z)∩Fix(Z)=Inn(C(Z)).But Z∈𝔐(Γ) and so C(Z)=Z. Thus, we have an extension(5.14)0→Inn(Z)→Fix(Z)→Out(W)→1which is easily seen to be
central.
Ifϕ∈Fix∘(Z) and α∈Aut(𝔉(Γ)), then α^ϕα^−1∈Fix∘(Z) and, because Inn(Z) is Abelian,(5.15)r(α^ϕα^−1)=r(ϕ).Since (5.12) splits, every element f∈Fix(Z) has a unique expression f=ϕ⋅α^, where ϕ∈Fix∘(Z) and α∈Aut(𝔉(Γ)). Consequently, we may define(5.16)r′:Fix(Z)→Inn(Z)by r′(f)=r(ϕ) for all f∈Fix(Z). If g∈Fix(Z) is written as ψ⋅β^,
then, by (5.15),(5.17)r′(fg)=r′(ϕα^ψα^−1αβ^)=r(ϕα^ψα^−1)=r(ϕ)r(α^ψα^−1)=r(ϕ)r(ψ)=r′(f)r′(g).Thus, r′ is a homomorphism and is easily verified to be
a retraction. The proof is completed in a manner similar to the conclusion of
the proof of Theorem 5.4, replacing Out∘(W) with Out(W), and so on.As an immediate
consequence we obtain the following corollary (cf. [6]).
Corollary 5.7.
IfW is a right-angled Coxeter group satisfying the
hypotheses of Theorem 5.6, then every extension(5.18)1→W→G→Q→1splits.Example 5.8.
The graph in Figure2 clearly satisfies the
hypothesis of Theorem 5.6. On the other hand, Theorem 5.6 does not apply to
either of the graphs given in Figure 1.Figure 2Remark 5.9.
As noted in the introduction, it is currently unknown whether, forW with trivial center, the sequence(5.19)1→Inn(W)→Aut(W)→Out(W)→1always splits. However, it was
recently shown in [8]
that sequence (1.5) always splits, but the splitting found there is not, in
general, compatible with (1.1). In particular, one cannot obtain generalizations
of Theorem 5.6 and Corollary 5.7 from [8].



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