(de Gruyter 2002
Rigidity of skew-angled Coxeter groups
Bernhard Mu¨hlherr and Richard Weidmann
(Communicated by R. Weiss)
Abstract.A Coxeter system is called skew-angled if its Coxeter matrix contains no entry equal to 2. In this paper we prove rigidity results for skew-angled Coxeter groups. As a consequence of our results we obtain that skew-angled Coxeter groups are rigid up to diagram twisting.
1 Introduction
Given a Coxeter matrixMover a setI, the corresponding Coxeter diagramGðMÞis the graph ðI;EðMÞÞwhereEðMÞdenotes the set of all 2-element subsetsfi;jg ofI such thatmij0 yand where each edge is labelled by the correspondingmij. We say that M is indecomposable if GðMÞ is connected; we say that M is 1-connected if GðMÞ is connected and if GðMÞ remains connected if one vertex is removed. We further say thatMis edge-connected ifMis 1-connected and ifGðMÞstays connected if the two vertices of an edge are removed. An edge of GðMÞis called a bridge, if it is not contained in a circuit ofGðMÞ. We say that a Coxeter system isskew-angledif the associated Coxeter matrix is skew-angled, i.e. contains no entry equal to 2.
LetðW;SÞbe a Coxeter system. Following [5] we call a setS0HW fundamental if ðW;S0Þis a Coxeter system. In [3] it is defined what it means for two fundamental sets to betwist equivalent, see Definition 4.4 of [3] or Section 7 below. It is in partic- ular very easy to decide whether for two given fundamental sets there exist twist equivalent sets that have isomorphic Coxeter graphs. Understanding the equivalence classes of fundamental sets of Coxeter groups would therefore solve the isomorphism problem. Our main result is the following:
Main Theorem. Suppose that ðW;SÞ is a skew-angled Coxeter system, let T¼SW denote the set of its reflections and suppose that SHT is a fundamental set.Then the following hold:
1. There exists a fundamental set S0HT that is twist equivalent to S and a bijection a:S!S0such thataextends to an automorphism of W.
2. If GðW;SÞhas no bridge,then one can choose S¼S0anda¼idS.
3. If S has at least 3 elements and if GðW;SÞ is edge-connected, then there exists wAW such that Sw¼S.
Reformulating statement (1) of the main theorem in the language of [3] we get the following corollary. It implies that Conjecture 8.1 in [3] holds in the skew-angled case.
Corollary A.Skew-angled Coxeter systems are reflection-rigid up to diagram twisting.
Remark 1. Let ðW;SÞbe a Coxeter system and let s;tASbe two reflections corre- sponding to the vertices which are on a bridge of GðW;SÞ. If there is a non-trivial reflection-preserving outer automorphism a of hs;ti (like for instance in the case where st has order 5), then it has an extension to a reflection-preserving auto- morphismb ofWandbðSÞis not twist equivalent toSbecause twistings are ‘angle- preserving’.
IfðW1;S1ÞandðW2;S2Þare both skew-angled Coxeter systems then any isomorphism f:W1!W2maps reflections onto reflections since the parabolic dihedral subgroups are the maximal finite subgroups and since any automorphism of a dihedral group D2n withnd3 maps reflections onto reflections. The theorem therefore gives a solu- tion to the isomorphism problem for the class of skew-angled Coxeter groups. This can be rephrased as follows:
Corollary B.Given two fundamental sets S;S0in a Coxeter group W such thatðW;SÞ andðW;S0Þare skew-angled,thenGðW;SÞandGðW;S0Þare twist equivalent.
If ðW;SÞis a skew-angled Coxeter system andRis a fundamental set of reflections, thenGðW;SÞ is twist equivalent toGðW;RÞby Corollary A. It is therefore easy to determine all Coxeter systems ðW0;S0Þ such thatW0 is isomorphic to Wif we can guarantee that each fundamental setRinWconsists of reflections. This motivates the definition ofreflection-independence. Following [1] we call a Coxeter groupreflection- independent ifRJSW for any two fundamental setsSandRofW. Our next result provides an easy criterion to see whether a skew-angled Coxeter group is reflection- independent. We call a vertex in a graphGanend-pointif it is contained in precisely one edge; an edge is called aspikeif it contains an endpoint.
Theorem (Reflection-independence criterion). Let ðW;SÞbe a skew-angled Coxeter system.Then W is reflection-independent if and only if there is no spike whose label is twice an odd number.
As there are no spikes in an edge-connected graph, Part 3 of the main theorem and the previous theorem have the following consequence:
Corollary C. Skew-angled Coxeter systems whose diagram has no spike which is la- belled by twice an odd number are rigid up to diagram twisting (in the sense of [3]);
they are strongly rigid (in the sense of [3]),if there are at least3 generators and the diagram is edge-connected.
Remark 2. Complete graphs on at least 3 vertices are edge-connected and hence one recovers a slight generalization of a result of A. Kaul in [14]. We have further learned that F. Haglund [12] has obtained a proof of our main theorem under the additional assumption that the Coxeter graph is a complete graph.
Remark 3.The theorem about reflection-independence follows from Proposition 9.4 below. This proposition can be used to deduce an algorithm to decide for an arbitrary Coxeter systemðW;SÞwhether there is a skew-angled fundamental setRofW.
In Section 2 we fix notation, we recall some definitions concerning the chamber system associated to a Coxeter system (i.e. its Cayley graph) and we deduce a crucial fact concerning roots and finite subgroups (cf. Lemma 2.6).
In Section 3 we consider reflections on thin chamber systems and we introduce geometric sets of rootsin a thin chamber system. This notion is motivated by a result of Tits in [21]. Much of the content of Section 3 is certainly known to the experts as it is closely related to the results of M. Dyer [8] and V. Deodhar [7] on subgroups generated by reflections in Coxeter groups; the setup which is used here is however more similar to the revision of these results due to J.-Y. He´e [13].
In Section 4 we investigate universal sets of reflections in a Coxeter system, i.e. sets of reflections which constitute a Coxeter system with the subgroupW0they generate.
Certain of these universal sets have the property that one can associate a root to each of its reflections such that the intersection of these roots is a fundamental domain for the subgroupW0; these are precisely the geometric sets of reflections.
In Section 5 we recall the definition of strong reflection-rigidity given in [3] and show that this definition is equivalent to a property of Coxeter systems which can be expressed by the notion of a geometric set of reflections.
In Section 6 we show that skew-angled Coxeter systems are strongly reflection-rigid if the underlying diagram is edge-connected by showing that they satisfy the equiva- lent definition given in Section 5. Here we use a special case of a result of R. Charney and M. Davis on rigidity of Coxeter groups (cf. [5]).
In Section 7 we use techniques introduced by M. Mihalik and S. Tschantz [15] to study how splittings of Coxeter groups over finite special subgroups behave with re- spect to di¤erent fundamental sets.
In Section 8 we give the proof of the main theorem by applying the results of Sec- tion 6 and Section 7.
In Section 9 we prove Proposition 9.4 which implies the reflection-independence criterion and which justifies Remark 3.
2 Preliminaries
Graphs. LetXbe a set, thenP2ðXÞdenotes the set of all subsets ofXhaving cardi- nality 2. A graph is a pair ðV;EÞconsisting of a set V and a set EJP2ðVÞ. The elements ofVandEare calledverticesandedgesrespectively.
LetG¼ ðV;EÞbe a graph and letWbe a subset ofV; thenGW denotes the graph ðW;P2ðWÞVEÞ.
LetG¼ ðV;EÞbe a graph. Letv;wbe two vertices ofG. They are called adjacent if fv;wgAE. Apath from vto wis a sequence v¼v0;v1;. . .;vk ¼w, where vi1 is adjacent to vi for all 1cick; the number kis the lengthof the path. Thedistance betweenvandw(denoted bydðv;wÞ) is the length of a shortest path joining them; if there is no path joiningvandw, we putdðv;wÞ ¼y.
A path v¼v0;v1;. . .;vk ¼w is said to be closed if v¼w; a closed path
v¼v0;. . .;vk¼v is called a circuit if v1;. . .;vk are pairwise distinct and if
kd2; a circuit v¼v0;v1;. . .;vk ¼v is called chordfree if EVP2ðfv1;. . .;vkgÞ ¼ ffv0;v1g;fv1;v2g;. . .;fvk1;vkgg.
The relation RHV V defined by R¼ fðv;wÞ jdðv;wÞ0 yg is an equivalence relation whose equivalence classes are called theconnected componentsofG. A graph is said to beconnectedif it has only one connected component.
Let vAV and letWbe the connected component of Gwhich contains v. Then v is called acut-pointofGifGWnfvgis not connected. The graphGis called 1-connected if it is connected and if there are no cut-points.
Let eAE be an edge of G and let W be the connected component of G which containse. Theneis called acut-edgeifGWneis not connected. The graphGis called edge-connectedif it is 1-connected and if there are no cut-edges.
We shall need the following facts about 1-connected graphs; the proof is straight- forward.
Lemma 2.1.LetG¼ ðV;EÞbe a1-connected graph with at least three vertices.Then i) every edge is contained in a chordfree circuit;
ii) given v;w;w0AV such thatfv;wg;fv;w0gAE and such that w0w0,then there ex- ists a sequence w¼w0;w1;. . .;wl¼w0 such that fv;wigAE, such that wi10wi and such that wi1;v;wiare in a chordfree circuit for1cicl.
Coxeter matrices, Coxeter diagrams and Coxeter systems. Let I be a finite set. A Coxeter matrix over Iis a symmetric matrixM¼ ðmijÞi;jAI with entries in NUfyg such thatmii¼1 for alliAIandmijd2 for alli0jAI. Given a Coxeter matrixM, we putEðMÞ:¼ ffi;jgHIj10mij0 yg. TheCoxeter diagram associated to Mis the graphðI;EðMÞÞwhose edges are labelled by the correspondingmij. The Coxeter matrix (and the associated Coxeter diagram) is calledindecomposableif the associated diagram is connected.
Theincidence diagram associated to a Coxeter matrix Mis the graphðI;E0ðMÞÞ where E0ðMÞ ¼ ffi;jgHIjmijd3g and where the edges are labelled by the corre- sponding mij. The Coxeter matrix (and the Coxeter diagram) is called irreducibleif the associated incidence diagram is connected.
Let Mbe a Coxeter matrix over I. A Coxeter systemof type Mis a pairðW;SÞ consisting of a groupWand a setS¼ fsijiAIgJW such thatSgeneratesWand such that the relationsððsisjÞmijÞi;jAIform a presentation ofW. Given a Coxeter system
ðW;SÞ, an element ofWis called areflectionif it is conjugate inWto an element of S; the set of all reflections is denoted byT.
LetMbe a Coxeter matrix overI. Given a subsetJofI,MJ denotes the restriction ofMontoJ. LetðW;SÞbe the Coxeter system of typeM. We putWJ ¼hsjjjAJi; it is a fact thatðWJ;fsjjjAJgÞis the Coxeter system of typeMJ. The groupsWJ are called the special subgroupsof the Coxeter systemðW;SÞ; aparabolic subgroupi s a subgroup which is conjugate to a special subgroup.
A Coxeter matrix (or diagram) is called spherical if the corresponding Coxeter group is finite. Given a Coxeter diagramMoverI, a subsetJofIis calledspherical ifMJ is spherical.
We close this subsection by recalling a well known fact [2].
Proposition 2.2.Let M be a Coxeter matrix over a set I and letðW;SÞbe the corre- sponding Coxeter system. Given a finite subgroup U of W, there exists a spherical subset J of I and an element wAW such that Uw is a subgroup of WJ.
The chamber system associated to a Coxeter system.LetðW;SÞbe a Coxeter system of typeMwhereMis a Coxeter matrix over a setI. The chamber system associated to ðW;SÞ is the graph C¼ ðC;PÞ, where C¼W and fc;dgAP if c1d AS. The vertices ofCare calledchambers, the edges are calledpanels. Two chambersc;d are calledi-adjacentifc1d ¼si. Since thesigenerateWthe graphCis connected. Note that we have a natural mappingtype:P!I, defined bytypeðfc;dgÞ ¼iifc1d ¼si. The groupWacts from the left (via left translation) onC. This action is regular on Candtype-preserving onP.
LetM;I;ðW;SÞandC¼ ðC;PÞbe as before, letJbe a subset ofIand letcAC.
The J-residueofcis the set RJðcÞ:¼cWJ. A residue is a subset ofC which is aJ- residue for some JJI. A residue is calledsphericalif it contains only finitely many chambers. In view of Proposition 2.2 and the regular action ofWonCthe following holds.
Lemma 2.3. A subgroup U of W is finite if and only if it stabilizes a spherical residue.
In view of the previous lemma the following proposition is a consequence of Proposition 5.5 in [6].
Proposition 2.4.LetðW;SÞbe a Coxeter system and let UcW be a finite subgroup of W.LetSU denote the set of all spherical residues stabilized by U and consider the graphGU whose set of vertices isSU and where two vertices are joined by an edge if one contains the other.ThenGU is connected.
Let M;I and ðW;SÞ be as before. Given a reflection tAW, we put PðtÞ:¼ fpAPjtp¼pgandCðtÞ:¼6
pAPðtÞp(soCðtÞis the collection of all cham- berscsuch thatcandtcare adjacent).
Note that for any wAW and tAT we have wPðtÞ ¼Pðwtw1Þ and wCðtÞ ¼ Cðwtw1Þ.
Lemma 2.5.LetðW;SÞbe a Coxeter system and let t be a reflection.Then the graph Ct¼ ðC;PnPðtÞÞhas two connected components.
Proof.This follows from Proposition 2.6 in [19]. r The two connected components ofCt are called theroots associated to t. Given a chambercand a reflectiont,Hðt;cÞ(resp.Hðt;cÞ) denotes the root associated tot, which containsc(resp. does not containc). Given a roota, the reflection to which it is associated is uniquely determined and it is denoted byra. Moreover, we denote by athe root which is associated toraand which is not equal toa. The set of roots will be denoted byFðW;SÞ.
Roots and spherical residues.LettAT be a reflection andAJCbe a residue. Thent stabilizesAif and only if both roots associated tothave non-trivial intersection with A. If this is not the case, then the unique root associated totwhich contains (resp.
does not contain)Awill be denoted byHðt;AÞ(resp.Hðt;AÞ).
Lemma 2.6. Let UcW be a finite subgroup of W and let tAW be a reflection such thatht;Uiis an infinite group.Then there exists a(unique)root associated to t which contains each spherical residue stabilized by U.
Proof. We define a graphGU as in Proposition 2.4 from which we know that it is connected. By Lemma 2.3 there exists a spherical residueAstabilized byU. Iftsta- bilizes A, then the groupV:¼ht;Ui stabilizesAandV is finite (again by Lemma 2.3). Therefore t does not stabilizeA which is equivalent to the fact thatAis con- tained in a root associated tot. This root will be calleda.
We have to show that any residue fixed byUis contained ina. LetBbe a spherical residue fixed by U. By the argument above it follows that Bis contained in a root associated tot. Suppose now thatBis not contained ina, i.e. thatBis contained in a. As the graphGU is connected (by Proposition 2.4) we can find a spherical residue B0that is stabilized byUsuch thatB0Va0 q 0B0Va. NowUandtstabilize the spherical residue B0and thereforeht;UistabilizesB0. It then follows from Lemma 2.3 thatht;Uiis finite. This yields a contradiction. r The root of the previous lemma will be denoted byHðt;UÞ. We also consider pairs of reflections t;t0 such thattt0 has infinite order. For those pairs denote the unique root associated to t which contains all spherical residues fixed by ht0i byHðt;t0Þ.
ThusHðt;t0Þ ¼Hðt;ht0iÞ.
The following observation is immediate:
Lemma 2.7.LetðW;SÞbe a Coxeter system,let t be a reflection and let U be a finite subgroup of W such that ht;Ui is infinite. Then each element wAW maps Hðt;UÞ onto Hðwtw1;wUw1Þ.
3 Reflections of thin chamber systems
Let G¼ ðV;EÞ be a graph and let I be a set. An I-labelling of G is a mapping t:E!Iwhose restriction to the set of edges through any given vertex is a bijection.
A thin chamber system over I is a pairC¼ ðG;tÞ consisting of a connected graph G¼ ðV;EÞand an I-labellingt of G. Throughout this section C¼ ðG;tÞ is a thin chamber system overI.
An automorphism of C is an automorphism of G which preserves the labelling;
thus an automorphism ofCfixing a vertex is already the identity. A reflection ofC is an elementrAAutðCÞsuch that the following conditions hold:
(1) r2 ¼id0rand
(2) ifErdenotes the set of fixed edges, then the graphðV;EnErÞhas two connected components.
Given a reflectionrofC, the two connected components of (2) are calledthe roots associated to rand the setCðrÞ:¼6
eAEreis called thewallassociated tor.
Aroot ofCis a subsetfofVsuch that there exists a reflection to whichfis asso- ciated as a root; this reflection is uniquely determined byf and it is denoted byrf. Given a rootfofC, thenf:¼Vnfis also a root. Given a reflectionrand a vertex v, letHðr;vÞdenote the root associated torwhich containsv. If v0 is another vertex then we callvandv0r-equivalentifHðr;vÞ ¼Hðr;v0Þ; in this case we writev@rv0.
In the following lemma we summarize some immediate observations:
Lemma 3.1. (1) Let r be a reflection and aAAutðCÞ. Then r0¼ara1 is a re- flection,aðCðrÞÞ ¼Cðr0ÞandaðfÞis a root associated to r0for each rootfassociated to r.
(2) Each root is convex, i.e. each path of minimal length between two vertices of a rootfis contained inf.In particular,intersections of roots are connected.
Let Xbe a subgroup of AutðCÞ and let ReflðXÞ denote the set of all reflections contained inX. Givenv;v0AVwe say thatvandv0areX-equivalentifv@rv0for all rAReflðXÞand we writev@Xv0in this case. IfvAV then a reflectionris called an X-wall ofvif rAReflðXÞand if there is a vertex v0@Xvwhich is inCðrÞ. The fol- lowing proposition is a direct consequence of Proposition 1 in [13]:
Proposition 3.2. Let X andReflðXÞbe as above and let v0 AV.Let R0 denote the set of X-walls of v0,let W:¼hR0iand let l:W !Nbe the length function with respect to the generating set R0.Then the following holds:
a) ReflðXÞJW andðW;R0Þis a Coxeter system;
b) for each rAR0 and each wAW one has lðrwÞ ¼lðwÞ þ1 if and only if wv0@rv0
and lðrwÞ ¼lðwÞ 1if and only if wv0Rrv0;
c) the action of W is sharply transitive on the X-equivalence classes of V.
We shall need also the following observation:
Lemma 3.3.The situation being as in the previous proposition let D:¼ fvAVjv@Xv0g.
Then D¼7
rAR0Hðr;v0Þ.
Proof.LetD0:¼7
rAR0Hðr;v0Þ. It is clear thatDJD0sinceD¼7
rAReflðXÞHðr;v0Þ.
LetvAD0nD. AsvAD0there exists a pathv0¼x0;. . .;xk¼vinD0joiningv0 andv by the second part of Lemma 3.1. Let 0<ickbe minimal for the property thatxiis not inD. It follows that there is a reflectionrAXswitchingxi1andxi. Asxi1isX- equivalent withv0 andxi1ACðrÞ, it follows thatrAR0 which impliesxiBD0. This contradicts the fact that the pathx0;. . .;xkis contained inD0. HenceD0nD¼qand
we are done. r
Given any group G acting on a set M, then we call FJM a prefundamental domainif gFVF0 q impliesg¼1; we call F a fundamental domain if it is a pre- fundamental domain and ifMis the union of thegF where gruns throughG.
We obtain the following consequence from Proposition 3.2:
Corollary 3.4. The situation (and notation) being as in Proposition 3.2 set
D:¼ fvAVjv@Xv0g. Then D¼7
rAR0Hðr;v0Þ and D is a fundamental domain for the action of W on V. Moreover, if rAR0 and wAW, then wDJHðr;v0Þ or wDJHðr;v0Þ; in the first case we have lðrwÞ ¼lðwÞ þ1, in the second we have lðrwÞ ¼lðwÞ 1.Finally,if r is a reflection in X,then there exists a reflection r0 in R0 which is W-conjugate to r.
A geometric pair of roots in G is a set of two roots f0f0 such that fVf0 i s a fundamental domain for the grouphrf;rf0i.
Lemma 3.5.Letf10f2 be a geometric pair of roots and let D¼f1Vf2.Put ri:¼rfi for i¼1;2 and X :¼hr1;r2i. Let l:X!Ndenote the length function with respect to the generating setfr1;r2g.Then D is an X-equivalence-class and the following holds for i¼1;2 and all xAX: xDJfi or xDJfi, lðrixÞ ¼lðxÞ þ1 if xDJfi and lðrixÞ ¼lðxÞ 1if xDJfi.
Proof. Let v0ADand let R0 be the set of X-walls of v0. Since the group Xis gen- erated by reflections it follows by Proposition 3.2 that ðX;R0Þ is a Coxeter system and that R0Jfr1;r2gX. We conclude that R0 has precisely two elements. Put D0¼7
rAR0Hðr;v0Þ; it follows from Lemma 3.3 thatD0¼ fvAVjv@Xv0g. There- foreD0JD. On the other handD0andDare fundamental domains for the action of XonVand therefore equality holds. HenceDi s anX-equivalence class. LetrAR0; as ri s an X-wall ofv0 there is an element vACðrÞVD0. ThusvACðrÞVDandrðvÞ is not in D0¼D; thus there exists aniAf1;2g such thatrðvÞAfi. As vAfi and vACðrÞwe conclude that r¼ri. This showsR0¼ fr1;r2g and the assertions of the
lemma follow now from Corollary 3.4. r
A set of rootsFis called 2-geometric, if each 2-element subset ofFis a geometric pair of roots; it is calledgeometricif it is 2-geometric and if7
fAFf0 q.
The definition of a geometric set of roots is motivated by the following proposition which is a consequence of Lemme 1 in [21] and the previous Lemma 3.5.
Proposition 3.6. Let F be a geometric set of roots. Let D¼7
fAFf, put R0¼ frfjfAFgand W ¼hR0i.ThenðW;R0Þis a Coxeter system and D is a funda- mental domain for the action of W on V.If v0AD,then R0is the set of W-walls of v0. Lemma 3.7.Letf10f2 be a geometric pair of roots.Iff10f2 is also a geometric pair of roots,then rf1commutes with rf2;in this casef10f2andf10f2are also geometric pairs.If D:¼ f1Vf20 q, then rf1 and rf2 generate a finite group and f10f2is a geometric pair as well.
Proof. Set r1¼rf1, r2¼rf2 andD¼f1Vf2. Let X¼hr1;r2iand letl:X !Nde- note the length function corresponding to the generating setfr1;r2g. Fori¼1;2 and eachxAX we havelðrixÞ ¼lðxÞ 1 if and only ifxDJfiandlðrixÞ ¼lðxÞ þ1 i f and only ifxDJfiby Lemma 3.3.
Suppose that f10f2 is a geometric pair of roots. Then f1Vf2, f1Vf2, r2ðf1Vf2Þ ¼r2f1Vf2 andr2ðf1Vf2Þ ¼r2ðf1ÞVf2 areX-equivalence classes which constitute a partition of V and henceX has 4 elements. Hence r1 commutes withr2andfr1;r2gis the set of reflections ofX. Thusf1Vf2,f1Vf2,f1Vf2 and f1Vf2 are the four X-equivalence classes of Vwhich shows that the four pairs involved are geometric.
Now suppose that D0¼ f1Vf20 q and choose vAD0. There exists xAX such that vAxD. It follows that xDJD0 and therefore lðr1xÞ ¼lðr2xÞ ¼lðxÞ 1 which implies that X is finite. Suppose that there is v0AD0 and let x0AX be the unique element such that v0Ax0D. Then lðr1x0Þ ¼lðr2x0Þ ¼lðx0Þ 1 and therefore x¼x0beause the longest element inXis unique. This shows thatD0¼xDis a fun- damental domain forX. Hencef10f2is a geometric pair. r
4 Universal and geometric sets of reflections
Throughout this section ðW0;S0Þ is a Coxeter system,T denotes the set of its re- flections,C¼ ðC;PÞis the associated chamber system andFdenotes the set of roots.
Note that the pairC¼ ðC;typeÞis a thin chamber system over Iin the sense of the previous section. Moreover, the elements of T are reflections of the thin chamber systemCin the sense of the previous section.
Let R be a subset ofT. Then we put MðRÞ ¼ ðoðrr0ÞÞr;r0AR where oðrr0Þdenotes the order ofrr0. The setRis calleduniversalifðhRi;RÞis a Coxeter system.
Let C be a set of roots. We put RðCÞ:¼ frcjcACg and MðCÞ:¼ ðoðrcrc0ÞÞc;c0AC.
A set Rof reflections will be called geometric(resp. 2-geometric) if there exists a geometric (resp. 2-geometric) set C of roots such that R¼RðCÞ; it will be called sharp-angledif each 2-element-subset ofRis geometric. We note that ‘sharp-angled’
is weaker than ‘2-geometric’.
The following observation is immediate from the definitions:
Lemma 4.1. Let R (resp. C) be a geometric set of reflections (resp. roots) and let R0JR (resp. C0JC). Then R0 (resp. C0) is a geometric set of reflections (resp.
roots).
Proposition 4.2. Let RJT be a set of reflections and let X ¼hRi.Then there exists a geometric set or reflections R0such thathR0i¼X and XVT¼R0X.
Proof.The groupXis generated by ReflðXÞand therefore we haveW¼X in Prop- osition 3.2 and Corollary 3.4. Let v0;D and R0 be as in Corollary 3.4 and put
C:¼ fHðr0;v0Þ jr0 AR0g. We claim thatR0 is geometric.
Let r00r00 AR0. We have to show thatf:¼Hðr0;v0Þand f0:¼Hðr00;v0Þconsti- tute a geometric pair. Obviously f0f0. Let Y be the group generated by r0 and r00 and let D0¼fVf0. Suppose that yðD0ÞVD00 q for some yAY. Ifl:W !N denotes the length function with respect to the Coxeter sytemðW;R0Þit follows that lðr0yÞ ¼lðr00yÞ ¼lðyÞ þ1. The length function on Y with respect to fr0;r00g is ob- tained by restricting l to Y because ðX;R0Þ is a Coxeter system by 3.2. Therefore it follows that yis the identity. This shows thatD0 is a prefundamental domain for the action of Y on C. As D0 contains a Y-equivalence class of C and as each Y- equivalence class is a fundamental domain by Corollary 3.4 (applied toY), it follows
thatD0is a fundamental domain forY. r
In the following three lemmas we summarize basic observations on subgroups gen- erated by 2 reflections; the first two of them are immediate consequences of Lemma 3.7.
Lemma 4.3. Let t0t0 be two reflections of the Coxeter system ðW;SÞ, and suppose tt0¼t0t.Ifa is a root associated to t and ifa0 is a root associated to t0,thenaVa0 is a fundamental domain forht;t0i.In particular the setft;t0gis geometric.
Lemma 4.4. Let t0t0 be two reflections of the Coxeter systemðW;SÞand suppose thatft;t0gis geometric and that tt0has finite order strictly greater than2.Ifais a root associated to t,then there is a unique roota0associated to t0 such thataVa0 is a fun- damental domain for the group ht;t0i;in this case the set aVa0 is a fundamental domain forht;t0ias well.
Lemma 4.5. Let t0t0 be two reflections and suppose that tt0has infinite order.Then Hðt;t0ÞVHðt0;tÞis a fundamental domain for the group ht;t0i.In particular,the set ft;t0g is geometric. Moreover we have Hðt;t0ÞJHðt0;tÞ, Hðt0;tÞJHðt;t0Þ and Hðt;t0ÞVHðt0;tÞ ¼q;in particular, Hðt;t0Þ0Hðt0;tÞis the only geometric pair of roots associated toft;t0g.
Proof. By Proposition 4.2 there exists a geometric set of reflections R0 such that X :¼ht;t0i¼hR0i. As X is an infinite dihedral group, it follows that R0 is X- conjugate to ft;t0g. Hence ft;t0g is a geometric set of reflections. Hence there is a geometric pair of roots f0f0 such that rf¼t and rf0 ¼t0; as tt0 has infinite
order it follows from Lemma 3.7 that fVf0¼q. As Hðt;t0ÞVHðt0;tÞ0 q 0 Hðt;t0ÞVHðt0;tÞ and Hðt0;tÞVHðt;t0Þ0 q we conclude that f¼Hðt;t0Þ and f0¼Hðt0;tÞ. The remaining assertions in the statement are now immediate. r Lemma 4.6.Let t;t0;t00be three pairwise distinct reflections of a Coxeter systemðW;SÞ and suppose that tt0;tt00 have infinite order and that Hðt;t0Þ0Hðt;t00Þ.Then t0t00 has infinite order.Moreover,if t0¼s0;s1;. . .;sk¼t00is a sequence of reflections with the property that si1si has finite order for 1cick, then tsl has finite order for some lAf1;. . .;k1g.
Proof. Suppose t0t00 has finite order. Then there is a spherical residue A stabilized by ht0;t00i; as A is stabilized by t0 and by t00 it follows that A is contained in Hðt;t0ÞVHðt;t00Þ. AsHðt;t0Þ ¼ Hðt;t00Þ, this intersection is empty and we have a contradiction. The second assertion is an immediate consequence of the first. r Lemma 4.7. Let RJT be a finite sharp-angled set of reflections,suppose that MðRÞ is irreducible and letC;C0be2-geometric sets of roots such that R¼RðCÞ ¼RðC0Þ.
a) If CVC00 q,thenC¼C0.
b) If CVC0¼q,thenC0¼ fcjcACgand rr0has finite order for all r;r0AR.
Proof.This is a consequence of Lemmas 4.4 and 4.5. r Lemma 4.8. Let RJT be a geometric set of reflections, suppose that MðRÞis irre- ducible and letC0C0 be geometric sets of roots such that R¼RðCÞ ¼RðC0Þ.Then MðRÞis spherical.
Proof. By Part b) of the previous lemma we haveC0¼ Cand as D¼7
cACc0 q 07
cACc¼D0we can find a chambercinD0. AsDis a fundamental domain forW ¼hRi, it follows that there existswAW such thatcAwðDÞ. By Proposition 3.6 Ris the set ofW-walls of any element in D; from Corollary 3.4 it follows that lðrcwÞ ¼lðwÞ 1 for all cAC where ldenotes the length function for the Coxeter systemðW;RÞ. This shows thatWis finite and the claim follows. r Proposition 4.9.Let R be a geometric subset of T. If MðRÞis non-spherical and irre- ducible then there is a unique geometric set of rootsCsuch that RðCÞ ¼R.
Proof.This is an immediate consequence of Lemma 4.8. r In the situation of the previous proposition the unique geometric set of roots will be denoted by CðRÞ; for each elementrAR, the unique root associated to rwhich i s i nCðRÞwill be denoted byHðr;RÞ.
Lemma 4.10. Let C¼C0Ufpg be a geometric set of roots such that MðCÞ is not spherical and irreducible and that MðC0Þ is spherical. Set t¼rp and
U¼hrc0jc0AC0i.Thenp¼Hðt;UÞ.In particular,if A is a spherical residue stabi- lized by the group U,then A is contained inp.
Proof. Let A be a spherical residue stabilized byU. If there is an infinity in MðCÞ then there is a rootc0AC0such thatrprc0 has infinite order. Asp0c0is a geometric pair of roots, it follows that p¼Hðrp;rc0Þ ¼Hðrp;AÞby Lemma 4.6. Thus we can assume that all entries inMðCÞare finite.
As C0 is a geometric set of roots and as the group UstabilizesAit follows that 7c0AC0c0VAis a fundamental domain for the action ofUonA; in particular, it is not empty. It follows therefore thatð7
c0AC0 c0ÞVAis not empty.
Suppose now that p¼ Hðt;UÞ. Then AJp and by the considerations above it follows that 7
c0AC0 c0Vp¼7
cACc is not empty. On the other hand fcjcACgis 2-geometric becauseCis geometric; hencefcjcACgis geometric.
As C is geometric andMðCÞis not spherical we obtain a contradiction to Lemma 4.8. Thusp¼Hðt;UÞandAis contained inHðt;UÞ. r
5 Strong reflection-rigidity
We recall the definition of a strongly reflection-rigid Coxeter system as it is given in [3]: A Coxeter systemðW;SÞis calledstrongly reflection-rigidif the following holds for each Coxeter system ðW0;S0Þ(whose set of reflections is denoted by T0): Given an isomorphisma:W !W0withaðSÞJT0, thenaðSÞisW0-conjugate toS0. We call a Coxeter diagram strongly reflection-rigid if the associated Coxeter system is strongly reflection-rigid.
Lemma 5.1. Let ðW;SÞ be a Coxeter system, let T denote the set of reflections, let RJT be universal and suppose that MðRÞis strongly reflection-rigid.Then R is geo- metric.
Proof.LetW0be the subgroup generated byRand letT0¼TVW0denote the set of reflections in W0. By Proposition 4.2 there is a geometric set of reflections R0 such that W0 is the group generated by R0 and such that T0 is the set of reflections of the Coxeter systemðW0;R0Þ. Now the identity onW0is an isomorphism mappingR onto a subset ofT0. As MðRÞis strongly reflection-rigid it follows that we can find w0AW0 such thatR0¼Rw0. This shows thatRis geometric. r We say that a Coxeter diagram M satisfies Condition (F) if the following is satisfied:
(F) Each universal setRof reflections in a Coxeter systemðW;SÞwithMðRÞ ¼M is geometric.
Proposition 5.2.A Coxeter diagram is strongly reflection-rigid if and only if it satisfies Condition(F).
Proof.Suppose thatMsatisfies condition (F), letðW;SÞbe a Coxeter system of type M, letðW0;S0Þbe an arbitrary Coxeter system whose set of reflections isT0and let a:W !W0be an isomorphism such thataðSÞJT0. ThenaðSÞis a universal sub- set of reflections inW0 and asMsatisfies Condition (F), it follows thataðSÞis a ge- ometric subset of T0. Let C0JFðW0;S0Þ be a geometric set of roots such that aðSÞ ¼RðC0Þ. According to Proposition 3.6 D0¼7
c0AC0c0 is a fundamental do- main for the action ofW0on its chamber system. This means thatD0consists of one chamber and hence aðSÞisW0-conjugate toS0. This shows one direction; the other
direction is provided by the previous lemma. r
Let ðW;SÞ be a Coxeter system and letT be the set of reflections ofðW;SÞ; we call a subsetRofTachordfree circuitif the Coxeter diagram associated toMðRÞis a chordfree circuit.
Lemma 5.3.LetðW;SÞbe a Coxeter system,let T denote the set of its reflections and let RJT be a universal set of reflections which is a chordfree circuit and which gen- erates an infinite group.Then R is geometric.
Proof. Since Ris a cordfree circuit and hRi is infinite it follows that hRi is a co- compact reflection group of the hyperbolic planeH2 or the Euclidean planeE2, see for example [23]. In particular hRiacts e¤ectively, properly and cocompactly on a contractible manifold. The main result of [5] then implies that the Coxeter system ðhRi;RÞis strongly reflection-rigid. ThusMðRÞis strongly reflection-rigid andRis
geometric by Proposition 5.2. r
6 The edge-connected case
Throughout this section we have the following setup:ðW;SÞis a Coxeter system,Tis the set of its reflections andRJT is a universal set of reflections such thatjRjd3, such that MðRÞ is skew-angled and such that the Coxeter diagram associated to MðRÞis edge-connected.
The goal of this section is to prove the following Theorem 6.1.The set R is geometric.
In view of Proposition 5.2 the previous theorem has following consequence.
Corollary 6.2. A skew-angled, edge-connected Coxeter diagram of rank at least 3 is strongly reflection-rigid.
Lemma 6.3. Given three pairwise distinct elements r;s;tAR,the product rsrt has infi- nite order.
Proof.If the order of all productsrs;rt;stis 3, then the three reflections generate the a‰ne Coxeter groupAA~2and the claim can be proved by considering the action of this
group on the Euclidean plane. In the remaining cases one uses the solution of the
word problem in Coxeter groups (cf. [20]). r
Since in the skew-angled case any chordfree circuit of a universal set generates an infinite group we have the following consequence of Lemma 5.3:
Lemma 6.4.Each chordfree circuit XJR is geometric.
It is further clear that for any chordfree circuitXJRthe Coxeter matrixMðXÞis irreducible. It follows that there exists a unique geometric set of rootsCðXÞsuch that X ¼RðCðXÞÞ(by Proposition 4.9). We recall that for each reflectionrAX the root which is contained inCðXÞand associated toris denoted byHðr;XÞ.
Lemma 6.5.The set R is sharp-angled.
Proof. Let r;sbe two distinct reflections in R. Ifrs has infinite order, then there is nothing to prove (by Lemma 4.5); ifrshas finite order, then, by Lemma 2.1 i), we can find a chordfree circuitXJRcontainingsandrasMðRÞis edge-connected. By the previous lemma we know that Xis geometric therefore fr;sgJX is geometric by
Lemma 4.1. r
Proposition 6.6.Let r;sAR be two distinct reflections such that rs has finite order and let C;C0be two chordfree circuits of R which contain r and s.Then Hðr;CÞ ¼Hðr;C0Þ.
Proof.LetC¼ fr¼t0;t1;. . .;tk ¼sgandC0¼ fr¼t00;t10;. . .;tl0¼sg.
Suppose first that t1 ¼t10. Then the group hr;s;t1i is infinite and the set X ¼ fr;s;t1g is geometric, because it is a subset of the geometric setC(cf. Lemmas 4.1 and 6.4). This showsHðr;CÞ ¼Hðr;XÞ ¼Hðr;C0Þ.
Suppose now thatt10t10. AsMðRÞis edge-connected we can find a sequencet1¼ s0;s1;. . .;sm¼t10 such that siBfr;sg and such that si1si has finite order for any 1cicm.
Assume now that Hðr;CÞ ¼ Hðr;C0Þ. As fHðr;CÞ;Hðs;CÞg and fHðr;C0Þ;Hðs;C0Þg are both geometric pairs of roots, it follows from Lemma 4.4 thatHðs;CÞ ¼ Hðs;C0Þ. Applying Lemma 4.10 twice it follows thatHðs;hr;t1iÞ ¼ Hðs;CÞ ¼ Hðs;C0Þ ¼ Hðs;hr;t10iÞ. Now we apply Lemma 2.7 with w:¼r to obtainHðrsr;hr;t1iÞ ¼ Hðrsr;hr;t10iÞ. By Lemma 6.3 we know thatrsrt1 andrsrt10 have both infinite order and therefore we obtain Hðrsr;t1Þ ¼ Hðrsr;t10Þ. By the second part of Lemma 4.6 there is an index j such thatrsrsj has finite order which yields a contradiction to Lemma 6.3.
HenceHðr;CÞ ¼Hðr;C0Þand we are done. r
Corollary 6.7. Given rAR and two chordfree circuits C;C0JR which contain r, we have Hðr;CÞ ¼Hðr;C0Þ.
Proof.By Proposition 6.6 the assertion is true if there is a reflectionsdi¤erent fromr which is contained inCandC0. If there is no such reflection we choosesAC,s0AC0
such thatsrandrs0 have finite order. Considering a sequences¼s0;. . .;sl¼s0 as in Lemma 2.1 ii), the claim follows by an obvious induction. r Given a reflection rAR, we define the root cr by choosing a chordfree circuit CJRwhich containsrand setting cr¼Hðr;CÞ, and we put C¼ fcrjrARg; the previous corollary ensures that the roots ðcrÞrARare well-defined, and as each edge of the diagramMðRÞis contained in a chordfree circuit (by Lemma 2.1 i)) we have the following.
Lemma 6.8. If r0sAR are such that rs has finite order,thenfcr;csg is a geometric pair of roots.
We will now prove the same result for two reflections inRwhose product has in- finite order:
Lemma 6.9.Let r;s;t;tAR be pairwise distinct reflections such that tr;ts;tthave finite order and such that sr has infinite order and suppose that there are two chordfree cir- cuits X;X0JR containingfr;t;tgandfs;t;tgrespectively.Then Hðr;sÞ ¼cr. Proof. LetX0¼ ft;t¼s00;s10;. . .;sl0¼sg. By Lemma 6.3 it follows thatstttandrttt have infinite order. As X is a chordfree circuit it follows from Lemma 4.10 that cr¼Hðr;XÞ ¼Hðr;ht;tiÞ ¼Hðr;tttÞ.
Suppose that cr¼ Hðr;sÞ. Then Hðr;tttÞ ¼ Hðr;sÞ. Setting sj ¼tsj0t for 0cjcl and slþ1¼s we have Hðr;s0Þ ¼Hðr;tttÞ ¼ Hðr;sÞ ¼Hðr;slþ1Þ and si1siis of finite order for 1ciclþ1. It follows by Lemma 4.6 thatrsk has finite order for somekAf1;. . .;lg. This contradicts Lemma 6.3 becausersk ¼rtsk0t. r Lemma 6.10.Let r;t;sAR be such that rt and st have finite order and such that rs has infinite order.Thencr¼Hðr;sÞ.
Proof.By Lemma 2.1 there exists a sequencer¼s0;. . .;sl¼ssuch thattsihas finite order, such that si10si and such that fsi1;t;sig is contained in some chordfree circuitXi for 1cicl. We choose such a sequence withlminimal.
Ifl¼1 we havecr¼Hðr;X1Þ ¼Hðr;sÞbecausesAX1 (cf. Lemma 4.10). Ifl¼2 the assertion follows by the previous lemma.
Suppose l>2. Then rsj has infinite order for 2cjcl by the minimality of l and cr¼Hðr;s2Þ by the previous lemma. Suppose now that Hðr;sÞ ¼ cr. Then there exists jAf3;. . .;lg such that Hðr;sj1Þ ¼ Hðr;sjÞ. Let Xj¼ ft;sj1¼s0;. . .;sk ¼sjg; by Lemma 4.6 there exists iAf1;. . .;kgsuch that rsi has finite order. ChoosingiAf1;. . .;kgmaximal for this property, we obtain a chordfree circuitft;r;si;siþ1;. . .;sk ¼sjgcontradicting the minimality ofl. r Proposition 6.11.Let s;rAR be such that sr has infinite order.Thencr¼Hðr;sÞand crVcsis a fundamental domain forhr;si.In particular,fcr;csgis geometric.
Proof.As the Coxeter diagram associated toMðRÞis connected, we have a sequence
r¼s0;. . .;sk¼sinRsuch thatsi1sihas finite order for 1cick; we choose a se-
quence with k minimal. The minimality of k implies that rsi has infinite order for 2cick. The previous lemma yieldscr¼Hðr;s2Þ. AsHðr;si1Þ ¼Hðr;hsi1;siiÞ ¼ Hðr;siÞfor 3cick(by Lemma 4.10) an obvious induction showsHðr;sÞ ¼cr.
In view of Lemma 4.5 the second assertion is an immediate consequence of the
first. r
Corollary 6.12.Let r0sAR be such that rs has finite order,let A be a spherical resi- due stabilized byhr;siand let tARnfr;sg.Then AJctand7
cACc0 q.
Proof. If rt andst have finite order, thenfr;s;tg is a chordfree circuit. Therefore it follows from Lemma 4.10 thatct¼Hðt;hr;siÞand henceAJct.
If rthas infinite order, thenct ¼Hðt;rÞby the previous proposition and asrsta- bilizesAit follows thatAJct. Ifsthas infinite order the same argument applies and the first assertion is proved.
AsAis a spherical residue stabilized byrandsit follows thatY ¼AVcrVcs0 q and asAJctfor alltARnfr;sgit follows thatq 0YJ 7
cACc. r
Proof of Theorem 6.1. It follows from Lemma 6.8 and Proposition 6.11 that the set C¼ fcrjrARgis 2-geometric. Moreover, by Corollary 6.12 we have7
rARcr0 q and thereforeCis geometric; asR¼RðCÞit follows thatRis geometric.
7 Visual decompositions and diagram twisting
We study decompositions of Coxeter groups as fundamental groups of graphs of groups. We therefore apply the ideas of M. Mihalik and S. Tschantz [15]. Suppose that ðW;SÞis a Coxeter system. Following [15] we call a splitting ofWas a funda- mental group of a graph of groups A visual (with respect to S) if every edge and vertex group isspecial, i.e. is generated by a subset ofS. It is clear thatAmust be a tree of groups since Wis generated by elements of finite order and therefore admits no non-trivial homomorphism toZ;Wcan therefore not be an HNN-extension. We use the following facts from [15], the second is actually a corollary of the first. Note that the definition of TS in Proposition 7.1 below makes sense since any sASi s of finite order and therefore acts with a fixed point. The fact thatWacts without fixed point guarantees its uniqueness.
Proposition 7.1(Mihalik, Tschantz).Suppose thatðW;SÞis a Coxeter group that acts simplicially without inversion on a simplicial tree T such that W fixes no vertex of T.
Then
W ¼p1ðAÞ
where A¼ ðTS;fGejeAETSg;fGvjvAVTSg;ffejeAETSgÞ is the graph of groups where the objects are defined as follows: TSHT is the unique minimal tree such that
for any sAS there exists xATS such that sx¼x.The vertex and edge groups are de- fined as Ge¼hsASjse¼eifor each edge e of TS and Gv¼hsASjsv¼vifor each vertex v of TS.All boundary monomorphisms are simply the inclusion maps.
Suppose now that a Coxeter group splits as a proper amalgamated product W ¼ACB. Then W acts on the associated Bass–Serre tree T. Since the amalga- mated product is proper it follows thatWacts without a fixed point. Proposition 7.1 therefore guarantees that Wsplits visually over a subgroup that fixes an edge ofT.
Since any edge stabilizer is conjugate toCwe have the following:
Theorem 7.2(Mihalik, Tschantz).LetðW;SÞbe a Coxeter group and suppose that W splits as a proper amalgamated product W ¼ACB.Then there exists a proper de- composition W ¼A0C0B0that is visual with respect to S such that C0is conjugate to a subgroup of C.
We recall the notion ofdiagram twistingas defined in [3]. Note that the operations we describe here are only a subset of the operations defined in [3], but that they co- incide when we restrict our attention to skew-angled Coxeter groups.
Suppose that a Coxeter group ðW;SÞsplits visually as an amalgamated product W ¼ACB where Cis a finite subgroup. This means that we have setsS1;S2HS such that A¼hS1i,B¼hS2iand C¼hS1VS2i. Letw be the longest element of the Coxeter groupðC;S1VS2Þ. This implies thatwðS1VS2Þw1¼S1VS2. It follows that
W ¼ACB¼AwCw1wBw1¼ACwBw1
where W¼ACwBw1 is visual with respect to the set S1UwS2w1 which is fun- damental. This is clear since wS2w1 is obviously fundamental forwBw1 and since S1VS2¼S1VwS2w1 is fundamental forC¼AVB¼AVwBw1. We say that the fundamental setsSandS¼S1UwS2w1areelementarily equivalent. We further say that fundamental setsSandSaretwist equivalentif there exists a finite sequence of fundamental sets S¼S1;S2;. . .;Sk1;S¼Sk such that Si and Siþ1 are elemen- tarily equivalent for 1cick. Note that we do not require the amalgamated product to be proper, i.e. possibly A¼C. This implies that conjugate fundamental sets are equivalent.
The names diagram twistingandtwist equivalent stem from the fact that the dia- grams GðW;SÞ and GðW;SÞ are related by a twisting operation. Namely both GðW;SÞand GðW;SÞ are obtained fromGðA;S1Þ and GðB;S2Þ by identifying the subdiagrams GðC;S1VS2Þ. In the first case the identification is the identity, in the second by the automorphism induced by conjugation with the longest element ofC.
Remark. Suppose thatðW;SÞ,W ¼ACBandS1 andS2 are as above. Instead of replacingS¼S1US2 byS1UwS2w1 we can replace it byw1S1wUS2. The result- ing diagram is clearly isomorphic to the first one since the two sets are conjugate.
Since any finite special subgroup is generated by either a subset ofS1 or ofS2 this
implies that any twisting operation on the diagram level can be realized such that any finite given spherical subset ofSis preserved.
This implies that the twist equivalences preserve angles, i.e. that if a fundamental setSis obtained from a fundamental setSofWthen any generating pairfs1;s2gHS of a finite dihedral group gets replaced with a pairfs1;s2gHSsuch thatfs1;s2gand fs1;s2gare conjugate inW.
The proof of the main theorem is by induction on #S, the cardinality ofS. We can assume thatGðW;SÞis not edge-connected otherwise the results follows from Section 6. In the case that GðW;SÞ is not edge-connected W decomposes visually as an amalgamated productACBwhereA,BandCare special subgroups ofðW;SÞand Cis either trivial or of order 2 or dihedral. In particular the Coxeter generating sets ofAandBare of smaller cardinality than S, i.e. we can assume that the respective results hold for each factor. We therefore need to study how a given visual splitting behaves with respect to another fundamental set.
Crucial to our arguments later is the following lemma which is a consequence of the work of V. V. Deodhar [6].
Lemma 7.3. Let ðW;SÞ be a Coxeter system and suppose that the decomposition W ¼G1C1G2C2G3is visual with respect to S.Suppose further that C1 is finite and that g2C2g12 ¼C1 for some g2AG2(possibly C1¼C2and g201).Then there exists a fundamental set S0 such that S is twist equivalent to S0 and such that the splitting W ¼G1C1G2g2C2g1
2 g2G3g12 ¼G1C1G2C1g2G3g12 is visual with respect to S0. Proof. By assumption C1 and C2 are special subgroups of the Coxeter group ðG2;S0Þ where S0¼SVG2. By Proposition 5.5 of [6] there exist sequences C1¼ U1;U2;. . .;Uk1;Uk ¼C2 andW1;. . .;Wk1 of finite special subgroups of ðG2;S0Þ such that Ui;Uiþ1HWi, such that Ui¼wiUiþ1w1i where wi is the longest element ofWi and such thatg2¼w1. . .wk1. This clearly implies that thek1 diagram
twists give the assertion of the lemma. r
Lemma 7.4.LetðW;SÞbe Coxeter system and C be a finite special subgroup such that W does not split over a proper subgroup of C. Then there exists a finite decomposition W ¼
CiAI
Gisuch that the following hold:
1. Gidoes not split over a subgroup of C for iAI.
2. W ¼
C iAIGiis visual with respect to a fundamental set S0that is twist equivalent to S.
Proof. We start with the trivial splitting, i.e. we putG1¼W. In particular we have a fundamental setS0¼Sthat is twist equivalent toSand a splittingW ¼
CiAI
Githat is visual with respect to this fundamental set.
If none of theGi splits over a subgroup ofCthere is nothing to show. If someGi
splits over a subgroup ofCwe show how to replaceS0by a twist equivalent set again denoted byS0and how to refine the splittingW¼
CiAI
Giby replacingGiby two new factors such that the obtained splitting is visual with respect to the new fundamental
set. Since Sis finite this process terminates and yields a decomposition and a setS0 with the desired properties.
Suppose that there existsiAI such thatGi splits over a subgroup ofC. IfGisplits over a proper subgroupC0ofCthenWalso splits overC0 since bothCandC0 are finite which contradicts our assumption. It follows thatGisplits overC. Theorem 7.2 implies thatGisplits visually over a subgroupC0 that is conjugate toC. Lemma 7.3 now guarantees that we can replaceS0with a equivalent set such thatC0¼C. This means that we can refine the visual splitting by replacingGiwith two factors. r Lemma 7.5.LetðW;SÞand C be as in Lemma7.4.Choose a set S0and a decomposi- tion W ¼
CiAI
Gi as in the conclusion of Lemma 7.4. Let SHT be a fundamental set of generators of W. Then there exists a fundamental set S0 that is equivalent to S such that the splitting W ¼
CiAI
Giis visual with respect to S0. Proof. We consider the amalgamated product W¼
CiAI
Gi as the graph of groups whose underlying graph has vertex set fxgUI, edge set f½x;i jiAIg, vertex groups Gi foriAI andGx¼C and all edge groups equal to C. The boundary monomor- phisms are the inclusion maps. We consider the action ofWon the Bass–Serre treeT with respect to this splitting.
We assume that among all sets that are equivalent to S the set S is the one such that the tree TS (of Proposition 7.1) has the smallest complexity, i.e. the least number of edges. We choose the associated graph of groups A¼ ðTS;fGejeAETSg;fGvjvAVTSg;ffejeAETSgÞ as in Proposition 7.1.
Claim 1.Ge¼StabWðeÞfor every edgeeAETS.
Proof. Suppose that Ge is a proper subgroup of StabWðeÞ for some eAETS. This implies thatWsplits over a proper subgroup of StabWðeÞ. Since the stabilizer of any edge is conjugate toCthis implies thatWsplits over a proper subgroup ofCwhich contradicts our assumption. This proves the claim.
Claim 2.IfvAVTS,e1;e2AETSande10e2thene1ande2 are notGv-equivalent.
Proof. Suppose thate1 ande2 areGvequivalent, i.e. that there exists agvAGvsuch that gve1¼e2. Since our ambient space is a tree we can clearly assume that both e1
ande2 are incident withv. Si nceGe1andGe2 are the full edge stabilizers ofe1 ande2
we have Ge1¼g1v Ge2gv. After collapsing all edges in the graph of groups except e1
and e2 we see that W splits as an amalgamated product W¼W1Ge1 W2Ge2W3. Because of Lemma 7.3 we can pass to an equivalent set S0 such that the splitting W ¼W1Ge1W2Ge2W3¼W1Ge1 W2g1v Ge
2gvg1v W3gv is visual with respect to S0. It is clear that the treeTS0 is contained in the treeT1Vg1v T2 where T1 is the component ofTSe2that containsvandT2is the component that does not contain v. (Note thatT1Vg1v T2is connected since both sets contain the terminal vertex ofe1
