Volume 2008, Article ID 976390,10pages doi:10.1155/2008/976390
Research Article
Automorphisms of Right-Angled Coxeter Groups
Mauricio Gutierrez1 and Anton Kaul2
1Department of Mathematics, Tufts University, Medford, MA 02155, USA
2Mathematics Department, California Polytechnic State University, San Luis Obispo, CA 93407, USA
Correspondence should be addressed to Anton Kaul,[email protected] Received 16 May 2008; Accepted 12 August 2008
Recommended by Alexander Rosa
IfW, S is a right-angled Coxeter system, then AutW is a semidirect product of the group Aut◦Wof symmetric automorphisms by the automorphism group of a certain groupoid. We show that, under mild conditions, Aut◦Wis a semidirect product of InnWby the quotient Out◦W Aut◦W/InnW. 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→InnW→ AutW→OutW→1 and consequently, all group extensions 1→W→G→Q→1 are trivial.
Copyrightq2008 M. Gutierrez and A. Kaul. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
A Coxeter group W is determined by its diagram Γ. It is known that in certain cases,W determinesΓas wellsee, e.g.,1,2. This is the case for right-angled Coxeter groups3,4, where the only relations ares21 for all generatorssandsttsfor some pairs of generators sand 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 endpointss and t means thatsandtcommute inW.
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 AutW. For example, the groupoid FΓconsisting of the vertex sets of complete subgraphs ofΓplays an important role in5where Tits exhibits a split exact sequence
1−→Aut◦W−→AutW−→Aut FΓ
−→1. 1.1
Tits also definesW to have propri´et´e I if the complementary graphΓC has no triangles. He goes on to show that ifWhas propri´et´e I, then Aut◦Wis isomorphic to InnW.
In the present article, we focus on the setMΓwhose members are the vertex sets of maximal complete subgraphs. We say thatΓhas condition C if there exist Z ∈ MΓand
a collectionT1, . . . , Tn∈MΓ− {Z}such that C1Ti∩Z /∅for 1≤i≤n,
C2for eachv∈Z,the cardinality of the set{i|v /∈Ti}is odd.
WhenΓhas condition C, the subgraph spanned byZ∪T1∪ · · · ∪Tnhas propri´et´e I. Thus, our condition C is a “local version” of Tits’ propri´et´e I.
Motivated by the results of Tits regarding sequence1.1, we consider the sequence
1−→InnW−→AutW−→OutW−→1. 1.2
A splitting of this sequence implies that all extensions of the form
1−→W−→G−→Q−→1 1.3
are trivial cf. 6. Clearly 1.2 does not always split. For example, Z2 is a right-angled Coxeter group and
0−→Z2−→Z4−→Z2−→0 1.4
is a nontrivial extension. On the other hand, if W has no center, then finding nontrivial extensions of W is surprisingly difficult. Whether1.2 splits for allW with trivial center is currently an open question.
The group Aut◦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.
aDoes the sequence
1−→InnW−→Aut◦W−→ Au t◦W/InnW −→1 1.5 split?
bAre the splittings of1.1and1.5compatible?
A positive answer to both a and b implies that1.2 is a split extension. We show in Theorem 5.4that ifΓhas condition C, then1.5splits. To obtain a splitting of1.2, we show that the action of AutFΓon Aut◦Wis compatible ifΓis not “too symmetrical.” More precisely, our main result is the following.
Theorem 1.1. IfΓhas condition C and eachα∈AutFΓleaves all vertices ofZinvariant, then 1.2is a split exact sequence.
It was recently shown8that1.5always splits. However, this result does not lead to a generalization ofTheorem 1.1as the splitting given there is not, in general, compatible with1.1.
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. See9or10for a comprehensive treatment.
IfXis any set, letP2Xdenote the set of subsets ofXwith cardinality 2.
Definition 2.1. Given a finite setSandE⊆ P2S,letΓ S, Edenote the undirected graph with vertex setSand edge setEnote thatΓdoes not have loops or parallel edges. As such graphs are often used to represent right-angled Coxeter groups,Γis called a Coxeter diagram.
Definition 2.2. GivenT ⊆ S,setET E∩P2T.The graphΓT T, ETis the subgraph ofΓ spanned byT.A complete subgraph is maximal if it is not properly contained in any complete subgraph ofΓ.
Definition 2.3. The presentation PΓ
S|s2,tu2;s∈S,{t, u} ∈E
2.1
is the Coxeter presentation defined byΓ.
Definition 2.4. A groupW is a right-angled Coxeter group if it has a presentationPΓdefined by some Coxeter diagramΓ.In this case, one writesWWΓand callsWthe right-angled Coxeter group defined byΓ.The pairW, Sis 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 2.6Radcliffe 3. IfW, SandW, Sare right-angled Coxeter systems forW,then there is an automorphismρ:W→Wsuch thatρS S.
A similar result was also obtained by Castella in4.
Definition 2.7. A subgroup ofWgenerated by a subset ofSis called a special subgroup. IfT ⊆S, it is customary to write WT for the subgroup generated byT. Finite special subgroups are called spherical subgroups. A subgroup ofWis 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, see9or10.
Theorem 2.8. IfA, B⊂S,thenWA∩WBWA∩B.
Theorem 2.9. IfT ⊆S,thenWT is the right-angled Coxeter group defined byΓT.
Corollary 2.10. The following are equivalent:
aWTis spherical;
bWTis an elementary abelian 2-group of rank|T|;
c ΓTis complete.
3. Automorphisms of right-angled Coxeter groups
For the remainder of this article, letW WΓbe the right-angled Coxeter group defined by the connected Coxeter diagramΓ S, E.It is easily verified that, under the operation of symmetric difference, the set
FΓ
T ⊆S|WTis finite
3.1 is a commutative groupoid with identity∅.
In5, Tits uses the group of automorphisms ofFΓto exhibit AutWas a semidirect product. We sketch the construction. Letσ ∈AutW.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 ofWis conjugate into a spherical subgroup. It follows that, for eachT ∈FΓ,there is a unique minimalT ∈FΓsuch thatσWTis conjugate intoWT.The map
q: AutW−→Aut FΓ
3.2 given byqσT T is an epimorphism.
Definition 3.1. Let Aut◦W be the kernel of q. Elements of Aut◦W are called symmetric automorphisms ofW.
Givenα∈AutFΓ,considerα∈AutWdefined by
αs
t∈αs
t 3.3
for alls∈S.A main result of5is the following theorem.
Theorem 3.2. The mapping AutFΓ→AutWgiven byα→αis a splitting of the sequence 1−→Aut◦W−→AutW−→Aut
FΓ
−→1. 3.4
LetdΓbe the standard path metric onΓthat assigns length one to each edge. For each s∈S,define
s∗
t∈S|dΓs, t≤1 , s⊥
t∈S|dΓs, t≥2
. 3.5
The subgraphΓs⊥spanned by the elements ofs⊥gives rise to certain generators of Aut◦W as follows. IfKis the vertex set of a connected component ofΓs⊥,then the mapσ :S →W given by
σt
⎧⎨
⎩
sts, t∈K,
t, t /∈K, 3.6
extends to a unique involutionσsK∈Aut◦W.The following is easily deduced from7.
Theorem 3.3. For each s ∈ S, let K1s, . . . , Ksms be the vertex sets of the components ofΓs⊥.Then Aut◦Wis generated by the set{σsKsi |s∈S, 1≤i≤ms}.
Remark 3.4. In7, M ¨uhlherr gives a complete presentation for Aut◦Wbased on a slightly different set of generators.
4. Symmetric automorphisms
The subsets of S that generate maximal spherical subgroups of W play a key role in the subsequent development. As such, we define
MΓ
T ⊆S|WTis a maximal finite subgroup
. 4.1
Note that MΓ 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◦Wand T ∈ MΓ,then there exists an element aT aTφ∈Wsuch thatφx aTxa−1T for allx∈WT.
Remark 4.1. When T ∈ MΓ, 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 cosetaTWT.
IfT, U ∈ MΓandT ∩U /∅,then aTxa−1T aUxa−1U for allx ∈ WT ∩WU WT∩U. Consequently,a−1T aUlies in the centralizer ofWT∩U.
Definition 4.2. Withφ, T, Uas above, letγTUγTUφ a−1T aU.A representative of the double cosetWTγTUWUis called aφ-transition fromWTtoWU.
Remark 4.3. IfT, U, V ∈ MΓ have pairwise nonempty intersection, thenγTT 1, γV U γUV−1,andγTUγUV 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∈MΓ.For much of what follows, Zmay be chosen arbitrarily; inSection 5we show that a preferred choice may exist.
Givenx ∈ W,let ψx be the inner automorphism of W given byw → xwx−1 for all w∈W.Restricting our attention toZwe define
InnZ
ψs∈InnW|s∈WZ
, Fix◦Z
φ∈Aut◦W|φx x∀x∈Z
. 4.2
In other words, Fix◦Zis the pointwise stabilizer ofWZunder the action of Aut◦Won W.
Observe that, since the subgraphΓZspanned byZis complete, InnZis Abelian.
Clearly, InnZ 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 classφ ∈ Out◦Wwith representative φ ∈ Aut◦W.Iff is the restriction ofφtoWZ,thenfx aZxa−1Z for allx ∈ WZ.Since f−1φ φandf−1φ∈Fix◦Z,it follows thatπis onto. We have established the following.
Theorem 4.4. The sequence
0−→InnZ−−→i Fix◦Z−−−→π Out◦W−→1 4.3 is a central extension.
Δ1 Ω Δ2
a
Δ1
Δ2 Ω Δ3
b Figure 1
In Section 5, we construct a retraction of the mapping InnZ →−i Fix◦Z. We now describe the key ingredient used in this construction. For eachT ∈FΓ,let
CT
v∈T
v∗. 4.4
It follows at once thatWCT∩Zis the centralizer ofWT∩ZinWcf.5, page 350. The function
πT :CT∩Z−→Z−T 4.5
given by
πTs
⎧⎨
⎩
1, s /∈Z−T,
s, s∈Z−T, 4.6
extends uniquely to a retractionWCT∩Z→WZ−T,which we also denote byπT. 5. Splittings
As inSection 4, we assume thatZis a fixed element ofMΓ.
Definition 5.1. One says that Z satisfies condition C if there exist elements T1, T2, . . . , Tn ∈ MΓ− {Z}such that the following conditions hold.
C1Ti∩Z /∅for each 1≤i≤n.
C2For eachv∈Z,the cardinality of the set{i|v /∈Ti}is odd.
IfΓcontains a maximal complete subgraphΩwhose vertex setZsatisfies condition C, then one says thatΓhas condition C. WhenΓhas condition C, then for each 1 ≤i≤nabove letΔi
denote the maximal complete subgraph ofΓspanned byTi.
Note that, sinceΓis connected, whenn 0 our hypotheses imply thatΓis complete.
In this case,Wis Abelian and the results below are trivial. Thus, we may assume thatnis a positive integer.
Example 5.2. We illustrate condition C with some examples.
aIfΓcontains a maximal complete subgraphΩwith an even number of vertices each of which meets exactly one other complete subgraph ofΓ,thenΓhas condition C.
For example, any Coxeter diagramΓthat containsFigure 1aas a subgraphwhere Ω,Δ1, andΔ2are maximal complete subgraphs ofΓhas condition C.
bSuppose n is odd and Γ has maximal complete subgraphs Ω,Δ1, . . . ,Δn. If the vertex set ofΩisZ {v1, . . . , vn}and the vertex set ofΔi contains every element of Z but vi, then Γ has condition C. For example, any Coxeter diagram Γ that containsFigure 1bas a subgraphwhereΩ,Δ1,Δ2,andΔ3are maximal complete subgraphs ofΓhas condition C.
Remark 5.3. In5, Tits says that a right-angled Coxeter group with Coxeter diagramΓhas
“propri´et´e I” if the complementary graph ΓC has no triangles. In this case, the inclusion InnW→Aut◦Wis an isomorphism. IfΓhas condition C, then the unionΔ1∪ · · · ∪Δn∪Ω defines a right-angled Coxeter group that has propri´et´e I. Thus, condition C is in some sense a “local” version of Tits’ propri´et´e I.
AssumeΓsatisfies condition C and letφ∈Fix◦Z.In this case, we can chooseaZφ 1.Then, for eachT ∈MΓwithT∩Z /∅,the transitionγZTφ aTφis an element of the cosetaTWT.It must be emphasized that no left multiplication by elements ofWZis permitted.
Theorem 5.4. IfΓhas condition C, then the sequence
1−→InnW−→Aut◦W−→Out◦W−→1 5.1
splits.
Proof. LetZ, T1, . . . , Tn∈MΓsatisfyingC1andC2ofDefinition 5.1. For eachφ∈Fix◦Z, define an inner automorphismrφ ψxwhere
x n
i1
πTi
γZTiφ n
i1
πTi
aTiφ
5.2
recall that ψx is the inner automorphism that conjugates each element of W by x. By definition, πTiγZTiφ πTiaTiφ lies in WZ−Ti ⊆ WZ. Consequently, the terms in the product5.2commute and so the mapping
r: Fix◦Z−→InnZ 5.3
is well defined. To see thatris a homomorphism, chooseφ, κ∈Fix◦Z.Then, for eachTiwe have
aTiφκ φ aTiκ
·aTiφ. 5.4
SinceaTiκlies in the centralizer ofWTi∩Z,any reduced expression foraTiκis of the form aTiκ v1· · ·vm,where eachvj∈CTi∩Z.Observe that
πTi φ
vj
⎧⎨
⎩
vj, ifvj∈Z−Ti, 1, ifvj/∈Z−Ti, πTi
vj
5.5
and soπTiφaTiκ πTiaTiκ.It follows from5.4that πTi
aTiφκ πTi
aTiκ
·πTi aTiφ
. 5.6
Now, using the fact that the image of eachπTi lies in the Abelian subgroupWZ,we have that rφκ rφrκ.
To see thatris a retraction, letv∈Zand letkbe a number ofTi’s that do not contain v.If ψv is the inner automorphism of W that conjugates byv,then aTiψv vfor every 1≤i≤n.From5.2we have thatrψvconjugates every element by
n i1
πTiaTiψv n i1
πTiv vk. 5.7
Sincekis odd,vkvand sorψv ψv.
Sinceris a retraction and the sequence4.3is central, the mapping
Fix◦Z−→InnZ×kerr 5.8
defined by
φ→rφ, rφ−1φ 5.9 is an isomorphism. Consequently,jπ|kerris an isomorphism from kerronto Out◦W.A section Out◦W→Aut◦Wis given by the compositionh◦j−1,whereh: kerr→Aut◦W is the inclusion
kerr⊆Fix◦Z⊆Aut◦W. 5.10
As noted in the introduction, to obtain a splitting of sequence1.2,Γmust satisfy an
“asymmetry” condition. This is obtained by imposing a restriction on the action of AutFΓ, and motivates the following definition.
Definition 5.5. Letα ∈ AutFΓandT ∈ MΓ.One says thatαfixesT ifα{t} {t} for everyt∈T.
Note that ifαfixesT,thenαT T for everyT ⊆T.
Theorem 5.6. IfΓhas condition C and eachα∈AutFΓfixesZ,then
1−→InnW−→AutW−→OutW−→1 5.11
splits.
Proof. Let FixZ {β∈AutW|βz zfor allz∈Z}.Since eachα∈AutFΓfixesZ,it follows thatαas defined in3.3abovelies in FixZand the mappingα→αis a splitting of the sequence
1−→Fix◦Z−→FixZ−−−→q AutFΓ−→1 5.12
where q is the restriction of the mapping q given in 3.2 above. As in the proof of Theorem 4.4, the projection FixZ→OutWis onto and its kernel is
InnZ∩FixZ Inn
CZ
. 5.13
Z
Figure 2
ButZ∈MΓand soCZ Z.Thus, we have an extension
0−→InnZ−→FixZ−→OutW−→1 5.14
which is easily seen to be central.
Ifφ∈Fix◦Zandα∈AutFΓ,thenαφ α−1∈Fix◦Zand, because InnZis Abelian, r
αφα−1
rφ. 5.15 Since5.12splits, every elementf ∈ FixZhas a unique expressionf φ·α, whereφ ∈ Fix◦Zandα∈AutFΓ.Consequently, we may define
r : FixZ−→InnZ 5.16
byrf rφfor allf ∈FixZ.Ifg∈FixZis written asψ·β, then, by 5.15, rfg r
φαψ α−1αβ r
φαψα−1 rφr
αψα−1 rφrψ rfrg.
5.17
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 ofTheorem 5.4, replacing Out◦Wwith OutW,and so on.
As an immediate consequence we obtain the following corollarycf.6.
Corollary 5.7. IfW is a right-angled Coxeter group satisfying the hypotheses ofTheorem 5.6, then every extension
1−→W−→G−→Q−→1 5.18
splits.
Example 5.8. The graph in Figure 2clearly satisfies the hypothesis of Theorem 5.6. On the other hand,Theorem 5.6does not apply to either of the graphs given inFigure 1.
Remark 5.9. As noted in the introduction, it is currently unknown whether, forWwith trivial center, the sequence
1−→InnW−→AutW−→OutW−→1 5.19
always splits. However, it was recently shown in8that sequence1.5always splits, but the splitting found there is not, in general, compatible with1.1. In particular, one cannot obtain generalizations ofTheorem 5.6andCorollary 5.7from8.
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