Rigidity of two-dimensional Coxeter groups

0.38 MB·English

by  Patrick Bahls

RIGIDITY OF TWO-DIMENSIONAL COXETER GROUPS
4
0
PATRICKBAHLS
0
2
Abstract. A Coxeter system (W,S) is called two-dimensional if the Davis
n
complex associated to (W,S) is two-dimensional (equivalently, every spher-
a
ical subgroup has rank less than or equal to 2). We prove that given a
J
two-dimensional system (W,S) and any other system (W,S′) which yields
1
thesamereflections,thediagramscorrespondingtothesesystemsareisomor-
2
phic,uptotheoperationofdiagramtwistingdefinedbyBrady,McCammond,

Mu¨hlherr, and Neumann in [8]. As a step in the proof of this result, certain
]
R two-dimensionalgroupsareshowntobereflectionrigidinthesenseof[8],and
a resultconcerning the strong rigidityof two dimensional systems is given in
G
thefinalsection.
.
h
t
a
m
1. Introduction
[

ACoxeter systemis apair(W,S)whereW isagroupwithapresentationofthe
2
v form hS | Ri, S ={si}i∈I, and
7
7
1 R={(sisj)mij|mij ∈{1,2,...,∞},mij =mji, and mij =1⇔i=j}.
1
1 When mij = ∞, the element sisj has infinite order. A group W with such a
3 presentation is called a Coxeter group, and S is called a fundamental generating
0 set.
/
h Let T ⊆ S. Denote by WT the subgroup of W generated by the elements in T.
t Suchasubgroupiscalledastandard parabolic subgroupofW,andanyconjugateof
a
m sucha groupis calleda parabolic subgroup. If WT is finite, WT is calleda spherical
subgroup. It is well-known (see [7], for instance) that (W ,T) is a Coxeter system
: T
v for any subset T ⊆ S, and therefore W is a Coxeter group in its own right, with
T
Xi theobviouspresentation. ItisalsoknownthatanysphericalsubgroupWT contains
a unique longest element with respect to the set S (see [7]), which we denote by
r
a ∆ . This element has the propertythat ∆ conjugates any element t∈T to some
T T
t′ ∈T.
The information contained in the presentation hS | Ri above can be displayed
nicely by means of a Coxeter diagram. The Coxeter diagram V associated to the
Coxeter system (W,S) is an edge-labeled graph whose vertices are in one-to-one
correspondence with the generating set S and for which there is an edge [s s ]
i j
labeled m between two vertices s and s if and only if i6=j and m <∞.
ij i j ij
GivenasphericalsubgroupW ofS,itisclearthatthesubgraphofV inducedby
T
the generatorsin T is a simplex in the combinatorialsense. We callsuch a simplex
2000 Mathematics Subject Classification. 20F28,20F55.
Key words and phrases. Coxetergroup,rigidity,diagramtwist.
Theauthor wassupportedbyanNSFVIGREpostdoctoral grant.
1
2 PATRICKBAHLS
a spherical simplex, and say that it is maximal if it is not properly contained in
another spherical simplex.
In the sequel, we frequently omit the word “Coxeter” when discussing groups,
systems, and diagrams, as these words will be used in no other context.
Itiseasytoseethatthediagramfullyandfaithfullyrecordsalloftheinformation
in the presentation hS | Ri. It is also easy to see that to a given group W there
may correspond more than one system (and therefore diagram). For example, the
dihedral group D of order 4k has the presentations
2k
ha,b | a2,b2,(ab)2k i
and
hc,d,g |c2,d2,g2,(cd)2,(cg)2,(dg)ki
when k is odd. These correspond to diagrams consisting of a single edge labeled
2k, and a triangle with edge labels {2,2,k},respectively.
Therefore one may consider the question: to what extent is a given Coxeter
system unique? As a first step toward answering this question, we must decide
what is meant by “unique”.
We say that the group W is rigid if given any two systems (W,S) and (W,S′),
there is an automorphism α ∈ Aut(W) satisfying α(S) = S′. Equivalently, the
diagramscorrespondingtothesetwosystemsareisomorphicasedge-labeledgraphs.
We say that W is strongly rigid if such an automorphism α can always be chosen
to lie in Inn(W); i.e., any two fundamental generating sets are conjugate to one
another.
We can relax these conditions slightly. We require the notion of a reflection. A
reflectioninthesystem(W,S)isanyconjugatewsw−1 ofagenerators∈S. Wesay
that a Coxeter system (W,S) is reflection rigid if given any other system (W,S′)
which yields the same reflections, there is an automorphism α of W satisfying
α(S) = S′. Finally, (W,S) is said to be strongly reflection rigid if given any other
system(W,S′)yieldingthesamereflections,suchanautomorphismαcanbefound
in Inn(W). We call W reflection independent if every two systems for W yield the
same reflections. Clearly if W is reflection independent, then (strong) rigidity and
(strong) reflection rigidity are equivalent.
A number of results have been proven that characterize the groups that satisfy
these rigidityconditions. Furthermore,there areothercharacterizationsofunique-
ness with which we will not concern ourselves in this paper. (See [1], [2], [3], [5],
[8], [11], [16], [17], [19], [20].)
In this paper we will generalize the method used in [3] in order to describe
the extent to which two-dimensional Coxeter groups are rigid. A system (W,S)
is called two-dimensional (or 2-d) if no three distinct generators from S generate
a finite subgroup of W. (The term “two-dimensional” refers to the dimension of
the Davis complex, a simplicial complex associated to the system (W,S). See [11],
[12] for more details regarding this complex and its usefulness.) The group W is
called two-dimensional if there exists a two-dimensionalsystem (W,S). (As a con-
sequenceofthemaintheorembelow,wewillseethatthisdistinctionisunnecessary
in the presence of reflection independence.) In order to describe the results we ob-
tain, we must introduce the important notion of diagram twisting, due to Brady,
McCammond, Mu¨hlherr, and Neumann (in [8]).
RIGIDITY OF TWO-DIMENSIONAL COXETER GROUPS 3
Given a Coxeter system (W,S), suppose that T and U are disjoint subsets of S
satisfying
1. W is spherical, and
U
2. every vertex in S \(T ∪U) which is connected to a vertex of T by an edge is
also connected to every vertex in U, by an edge labeled 2.
Undertheseconditions,wemaydefineanewdiagram(andthereforenewsystem)
V′ for W by changing every edge [tu] (t ∈ T, u ∈ U) to an edge [tu′], where
u′ =∆−1u∆ , leaving every other edge unchanged. This modification results in a
U U
generating set S′ obtained from S by replacing t∈T with ∆−1t∆ .
U U
This operation is called a diagram twist, because of the way that we “twist”
around the subdiagram representing the group W .
U
Werequireafewnewtermsinordertostatethispaper’smainresults,remaining
consistent with the terminology of [19]. If V is connected and s is a vertex in V
such that V \{s} is disconnected, s is called a cut vertex of V. If V has no such
vertices, we say that V is one-connected. If V is one-connected and there exists no
edge [st] such that V \[st] is disconnected, then V is called edge-connected. If V
is one-connected and there exists no edge [st] with odd label such that V \[st] is
disconnected, we call V odd-edge-connected. (Thus V is odd-edge-connected if it is
edge-connected.)
Theorem 1.1. Let (W,S) be a two-dimensional Coxeter system with diagram V.
Then (W,S) is reflection rigid, up to diagram twisting. (That is, given a system
(W,S′) which yields the same reflections as (W,S), there is a sequence of diagram
twists which transforms the first system into the second.)
As a step in the proof of the main theorem, we will prove
Theorem 1.2. Let (W,S) be a two-dimensional Coxeter system with odd-edge-
connected diagram V. Then (W,S) is reflection rigid.
Furthermore,wewillproveatheorem(Theorem6.2)concerningthestrongrigid-
ity of 2-d Coxeter groups. Its statement will be deferred until the final section of
this paper.
The above results partially generalize the similar results obtained by Mu¨hlherr
and Weidmann in [19]. Indeed, in this paper we will make similar use of the
results of [18] in order to complete the proof of Theorem 1.1 (see [19], Section
8). However, the preliminary arguments are very different, and will be introduced
in the following section. The author has also recently learned that T. Hosaka has
provenindependentlyaslightlyweakerresultconcerningrigidityoftwo-dimensional
Coxeter groups.
This is the third paper in a series (see [3], [4]) which makes use of similar tech-
niques inorderto establishstructuralpropertiesofCoxetergroups. It is clearthat
thesetechniquescanbepushedevenfurthertoproveresultsaboutyetmoregeneral
Coxeter groups. This will be done in subsequent papers.
The authorgratefully acknowledgeshelpful discussionswith Ruth Charney,Ilya
Kapovich, and Richard Weidmann during the writing of this paper.
4 PATRICKBAHLS
2. Circuits and centralizers
We begin by sketching the argument that we will use to prove Theorem 1.1.
Let (W,S) be a 2-d system, and let (W,S′) be another system for W. Denote the
correspondingdiagrams by V and V′. Let us assume until further mention that W
is reflection independent. Our goal is to show that, up to twisting, V and V′ are
identical.
The two-dimensionalityof (W,S) allows us to establish a matching between the
edges of V and the edges of V′, using the following result from [16].
Theorem 2.1. Let W be a Coxeter group with diagram V, all of whose maximal
spherical simplices are of the same dimension. Then, given any other Coxeter
system (W,S′) with diagram V′, there is a one-to-one correspondence φ between
the maximal spherical simplices of V and those of V′. Moreover, for any maximal
spherical simplex σ in V, there is an element w ∈W such that wW w−1 =W .
σ φ(σ)
In our case, every maximal spherical simplex is an edge, and therefore has di-
mension 1. We apply Theorem 2.1 to obtain a matching between the edges of V
and the edges of V′ which respects conjugacy as indicated in the theorem.
Why must each edge of V be matched with an edge of V′? If there were an edge
[st] in V such that wW[st]w−1 =Wσ for some σ of dimension >1, then Dn ∼=Wσ,
where n is the orderof st. However,it is an easy matter (see [1]) to show that this
can only happen if n = 2k, k odd, and σ is a triangle with edge labels {2,2,k}.
In this case, the central element of W (which is of even length with respect to
[st]
(W,S),andis thereforenota reflection)is areflectionin (W,S′), contradictingthe
assumptionthatW isreflectionindependent. Animmediatecorollaryisthatevery
system corresponding to W is 2-d, so it matters not whether we refer to the group
or to the system as 2-d, provided W is reflection independent. (Alternately, one
may apply Lemma 1.5 of [11].)
Let φ be the matching whose existence is guaranteed by Theorem 2.1.
We now consider circuits in the diagram V. A simple circuit of length k in V is
a collection C of k distinct edges {[s s ],...,[s s ]} for which s 6= s when i 6= j.
1 2 k 1 i j
Define d(i,j)=min{|i−j|,k−|i−j|} for 1≤i,j ≤k. We call a simple circuit C
achordal if for any two vertices s 6= s in C such that d(i,j) > 1, [s s ] is not an
i j i j
edge in V.
We shall prove the following theorem.
Theorem 2.2. Let (W,S), (W,S′), V, V′, and φ be as above. Let C be an
achordal circuit of length k in V, as above. Then there is an achordal circuit C′ =
{[sˆ sˆ ],...,[sˆ sˆ ]} in V′ such that {sˆ,sˆ } = φ({s ,s }) for 1 ≤ i ≤ k. More-
1 2 k 1 i i+1 i i+1
over, foreachedge[s s ]thereisanelementw ∈W suchthatw s w−1 =sˆ
i i+1 i+1 i+1 i i+1 i
and w s w−1 =sˆ both hold.
i+1 i+1 i+1 i+1
Thereforenotonly do the edgesmatchup nicely, butthe achordalcircuits do as
well. In fact, we can do better:
Theorem 2.3. Let (W,S), (W,S′), V, V′, φ, C, and C′ be as in Theorem 2.2. Let
s and s be distinct vertices on C, with {e =[s s ],[s s ],...,e =[s s ]}
i j i i i+1 i+1 i+2 j j−1 j
a subpath of C between them. Let w and w be the group elements which conjugate
i j
the edges e and e , respectively, to their corresponding edges in C′. Then w w−1
i j i j
can be written α α ···α , where for every l (1 ≤ l ≤ r) α ∈ S′ and one of the
1 2 r l
following holds.
RIGIDITY OF TWO-DIMENSIONAL COXETER GROUPS 5
1. For every l′ (1 ≤ l′ ≤ k), αl 6= sˆl′, and αl commutes with at least 2 vertices
which lie on C′. Moreover, we can find two such elements, sˆ and sˆ , such that
l1 l2
the path {[sˆ α ],[αsˆ ]} separates C′ into two circuits, one containing sˆ and the
l1 l l l2 i
other containing sˆ .
j
2. α = sˆ, in which case both [sˆ sˆ] and [sˆsˆ ] are labeled 2, or α = sˆ , in
l i i−1 i i i−1 l j
which case both [sˆ sˆ ] and [sˆ sˆ ] are labeled 2.
j−1 j j j+1
Although Theorem 2.3 appears very technical, it addresses precisely the issues
thatmustbefacedwhendealingwithstrongrigidityinthepresenceofedgeslabeled
2. (Compare the arguments of Section 6 in [3]; in particular, those used in Cases 1
and 2.) We note that in case no edges in V are labeled 2, w = w must hold for
i j
all edges w and w ; thus the circuit C is in this case “strongly rigid”.
i j
Mu¨hlherr and Weidmann also consider achordal circuits in [19], but their ap-
proach to these circuits is very different from that adopted here, where we draw
upon the techniques developed in [3] and [4].
OnceTheorem2.2andTheorem2.3havebeenestablished,itwillbearelatively
straightforwardmattertoreconstructtheunique (uptotwisting)diagramV which
is built up from the achordalcircuits.
As willbecomeclear,ouranalysisoftheachordalcircuitsinV willdependupon
an understanding of the centralizer C(s) of an arbitrary generator s ∈ S. To that
end,werecallinthenexttheoremthestructureofC(s)(firstgivenin[9]). We also
introduce notationwhich will remainfixed throughoutthe remainder of the paper.
Let (W,S) be an arbitrary Coxeter system and suppose s,t∈S are elements of
the fundamental generating set S. If m = 2k is even, denote by u the element
st st
(st)k−1s. We note that u commutes with t (in fact, u = s if st = ts). If
st st
m =2k+1 is odd, denote by v the element (st)k. Note that v sv−1 =t. More
st st st st
generally, there is an path in the diagram V between two vertices s and t which
consists entirely of odd edges if and only if s and t are conjugate to one another.
In fact, if {[ss ],[s s ],...,[s t]} is such a path, then v v ···v conjugates
1 1 2 k skt sk−1sk ss1
s to t.
Let V¯ be the graph resulting from a diagramV by removing all edges with even
labels. Asin[9],wecanidentifyelementsofthefundamentalgroupofV¯ withpaths
in V¯ which start and end at a fixed vertex s ∈ S and which never backtrack. For
the fixed vertex s∈S, let B(s) be a collection of simple circuits in V¯ containing s
such that B(s) generates the fundmental group of V¯.
The following was first proven by Brink in [9]. (The generators given here can
be computed by arguments similar to those in [6].)
Theorem 2.4. Let (W,S) be an arbitrary Coxeter system with diagram V, and let
s∈S. Then C(s) is the subgroup of W generated by
{s}∪A∪B
where
A={vu v−1 | v =v v ···v ,t,s ∈S,m even;m ,m odd}
tsk s1s s2s1 sksk−1 i tsk s1s sisi−1
and
B ={v v ···v | {[ss ],...,[s s]}∈B(s)}.
s1s s2s1 ssk 1 k
6 PATRICKBAHLS
We will use this description of the centralizer C(s) in the sequel.
Remarks. When(W,S)is2-d,itcanbeshownthatdistinctchoicesofs ,s ,...,s
1 2 k
andtinAandBaboveyielddistinctgenerators. Thismaynotbethecaseif(W,S)
is not 2-d.
Moreover, it is not difficult to compute a geodesic form for an element w in
C(s). To do this, first express w as a product in the given generators. Fac-
tor all occurrences of s as a single letter to the end of of the word and cancel,
yielding either s or 1. Next, perform all “obvious” cancellation; that is, given
α=vs1s···vssk and β =vs′1s···vss′k′ in B, for some i, 0≤i≤min{k,k′} we have
sk =s′1,sk−1 =s′2,...,sk−i+1 =s′i,sothatα·β =vs1s···vsk−i+1sk−1vs′i+1s′i···vss′k′.
SimilarcancellationoccursinaproductoftwogeneratorsfromA,andinaproduct
of a generator from A and a generator from B.
We claim that the wordthat results after suchcancellationis geodesic. In order
toprovethis,weappealtoaresultofTits. FromSection2of[21]weconcludethat
if the word resulting from the previous paragraph were not geodesic, we would be
able to shorten the word by successively replacing subwords (st)n with (ts)n when
st has order 2n and subwords (st)ns with (ts)nt when st has order 2n+1, and
then canceling any adjacent occurrences of the same letter which might arise in
the course of these replacements. However, thanks to two-dimensionality, no such
shortening replacements can be performed, (perhaps) aside from commuting the
single occurrence of s that may occur at the end.
The aboveargument(replacingtheonehalfofarelatorwiththe otherhalf)will
be used again in the following sections. We refer to the process of shortening a
word w in the manner described above as the Tits process (TP).
3. Matching edges in a given circuit
Inthis sectionweretracethe argumentsfrom[3],adaptingthem asnecessaryto
the case of 2-d systems. In fact, many of the arguments throughout the remainder
of the paper will parallel arguments from [3] (such analogous arguments will be
indicated).
Let (W,S) be a 2-d system with diagram V, and let (W,S′) be another system,
withdiagramV′, yieldingthesamereflectionsas(W,S). Wefix allofthis notation
for the remainder of the paper.
Let C = {[s s ],...,[s s ]} be an achordal circuit in V. By Theorem 2.1, for
1 2 k 1
every i = 1,...,k there exists an edge [s′′ s′] in V′ and an element w ∈ W such
i−i i i
that wiW[si−1,si]wi−1 = W[s′i′−1,s′i]. By considering the possible generators for the
dihedral group W[s′i′−1,s′i], we can assume that
w s w−1 =s′′ and w s w−1 =z s′z−1 ,
i i−1 i i−1 i i i i−1,i i i−1,i
for some word zi−1,i ∈ W[s′i′−1,s′i]. Let m = mi−1,i be the order of si−1si. One
may prove by direct computation that after suitably modifying w we can assume
i
z =(s′s′′ )j, where 0≤j ≤[m−1] if m is even and 0≤j ≤ m−3 if m is odd.
i−1,i i i−1 4 2
(Cf. [3], Section 4.) In particular, z =1 if m∈{2,3,4}.
i−1,i
We now use the fact that each vertex s appears in two edges in C. Because
i
s′ =z−1 w s w−1z and s′′ =w s w−1, both s′ and s′′ are conjugate to s ,
i i−1,i i i i i−1,i i i+1 i i+1 i i i
and therefore to each other. Let P = {[s′s ],...,[s s′′]} be a path of minimal
i i i,1 i,r i
length from s′ to s′′, all of whose edges have odd labels. Then the element
i i
RIGIDITY OF TWO-DIMENSIONAL COXETER GROUPS 7
v¯i =vsi,1s′ivsi,2si,1···vs′i′si,r
conjugates s′′ to s′.
i i
We now compute:
w s w−1 =z s′z−1 =z v¯s′′v¯−1z−1 =z v¯w s w−1v¯−1z−1 .
i i i i−1,i i i−1,i i−1,i i i i i−1,i i−1,i i i+1 i i+1 i i−1,i
Thus
w−1z v¯w ∈C(s )=C(w−1z s′z−1 w )=w−1z C(s′)z−1 w .
i i−1,i i i+1 i i i−1,i i i−1,i i i i−1,i i i−1,i i
Finally, we obtain
w w−1 =v¯−1s¯z−1 (1)
i+1 i i i i−1,i
for some s¯ ∈ C(s′). Denote by x the word appearing on the right-hand side of
i i i
(1). Then
x x ···x x =w w−1w w−1 ···w w−1 =1.
k k−1 2 1 1 k k k−1 2 1
Ifwechooseageodesicrepresentationofs¯,eachofthe wordsv¯−1,s¯,andz−1
i i i i−1,i
are as short as possible.
Recalling the generators of C(s′) given by Theorem 2.4, s¯ may terminate with
i i
a term of the form us′i′−1s′i (if m=ms′i′−1s′i is even) or vs′is′i′−1 (if m is odd). In this
case, we can reduce the product appearing in (1) by multiplying this term with
z−1 if z 6=1.
i−1,i i−1,i
There is one other place where reduction can occur in x . Suppose s¯ begins
i i
with a word of the form
vα1s′ivα2α1···vαlαl−1vs′iαl
corresponding to an odd “loop” based at s′ in the diagram V′. If
i
v¯i =vsi,1s′ivsi,2si,1···vs′i′si,r 6=1,
we may have
s =α,s =α ,...,s =α ,
i,1 l i,2 l−1 i,j l−j+1
where j ≤r. Therefore after cancellation v¯−1s¯ begins with a word of the form
i i
vβ1s′i′vβ2β1···vs′iβj (2)
for some j ≥0. (When j =0, (2) has the form vs′s′′.)
i i
After all cancellation has been performed on x , we obtain a new product, of
i
“even” words E (involving terms uss′) and “odd” words O (involving terms vst):
i
xi =vα1s′i′vα2α1···vs′iαjE1O1E2O2···ElOlw(s′i,s′i′−1) (3)
where w(s′,s′′ ) is some word in the letters s′ and s′′ . (We allow E and O
i i−1 i i−1 1 l
to be trivial.) If s′ appears in s¯ as a generator of C(s′), it may be absorbed by
i i i
w(s′,s′′ ). TheexactstructureofthewordsE andO isgovernedbyTheorem2.4.
i i−1
All of the terms vst, and all of the even terms uss′ which consist of more than a
i
single letter s will be called long terms. In case uss′ is a long term of length 2r+1
i
8 PATRICKBAHLS
(for r ≥ 1), we will also call the words (ss′)r and (s′s)r long terms by a slight
i i
abuse of terminology. Terms consisting of a single letter (either s′ or s such that
i
ss′ =s′s) will be called short terms.
i i
We claim that the product given in (3) is in fact geodesic; this is shown by
another application of the result of Tits. Thanks to the form of the long terms
and two-dimensionality of W, the only possible subwords in the right-hand side of
(3) which admit replacement as in TP would come from s¯. (E.g., TP may allow
i
us to bring two instances of s′ together, in order to cancel them.) But we have
i
assumeds¯ to be geodesic,and therefore unchangedunder applicationof TP (such
i
cancellations have already been performed).
We are nowready to begin the proofofTheorem2.2, inducting upon the length
k of the circuit C.
4. The base cases
We continue to use the notation from the previous sections, and prove Theo-
rem 2.2 and Theorem 2.3 for cycles of lengths 3 and 4. Some of the methods used
in this section will be generalized in the following section, and so will be stated in
general terms. We will use the fact that x x ···x = 1 in order to show that
k k−1 1
each word x must have a very specific form. The form of x will allow us both to
i i
identify a circuit in V′ to which C corresponds as in Theorem2.2 and to prove the
statements regarding w w−1 made in Theorem 2.3.
i j
Hereafter we say that the words x 6= 1 and x 6= 1 (i > j) are adjacent in the
i j
product x ···x if either i=j+1 or x =1 for j <l <i.
k 1 l
We first assume there is no cancellation of common short terms between two
words x and x . Having proven the theorems in this case, we will then indicate
i j
how to prove the theorems in general. (In fact, by two-dimensionality, the case
in which k = 3 yields very little such short term cancellation, as there can be no
vertex α not on C′ such that α commutes with two distinct elements s′ and s′.)
i j
Letusfirstconsiderthecaseofacircuitoflength3: x x x =1. Unlessallthree
3 2 1
wordsx aretrivial(inwhichcasew =w =w alreadyandC clearlycorresponds
i 1 2 3
to a circuit C′ in V′), at least two of these words are nontrivial.
Case 1. Suppose that x = 1 (after renaming, if necessary). Thus x x = 1 and
1 3 2
s′ =s′′.
1 1
First suppose that x3 ends with the long even term α(s′3α)m2−1 and x2 begins
with the long even term (βs′2)n2−1β. (If x2 were not to begin with an even term,
at most one pair of letters would cancel, and an application of TP would yield a
contradiction.) This implies that s′ = s′′. Since s′ 6= s′′ = s′, we can avoid the
2 2 3 2 2
same contradiction only if α=s′ and β =s′, in which case m=n and
2 3
α(s′3α)m2−1·(βs′2)n2−1β =s′3s′2.
In this case, easy computations (and applications of TP) show that there is no
further cancellation if us′s′ is either the first term in x3 or is preceded by another
2 3
long term, and if us′s′ is either the last term in x2 or is followed by another long
3 2
term. Therefore, in order that x3x2 = 1, us′2s′3 must be preceded by s′3 in x3, and
us′3s′2 must be followed by s′2 in x2. Moreover, if there are any further terms in x3
andx2,therecanbe nofurthercancellation. Thusx2 =x3 =(s′2s′3)m2 ,ands′i =s′i′
RIGIDITY OF TWO-DIMENSIONAL COXETER GROUPS 9
for i=1,2,3. Moreover, x1 = 1⇒w1 =w2, and (s′2s′3)m2 commutes with both s′2
and s′. Therefore, w s w−1 =s′, w s w−1 =s′, and
3 1 1 1 1 1 2 1 2
w1−1s′3w1 =w3−1(s′2s′3)m2 s′3(s′2s′3)m2 w3 =w3−1s′3w3 =s3.
Thus the same element (namely, w ) conjugates each s to sˆ = s′, proving Theo-
1 i i i
rem 2.3 for C.
Now suppose x3 ends with a long odd term (s′3α)m2−1 and x2 begins with a long
odd term (βs′2′)n−21. (As before, we obtain a contradiction to x3x2 =1 if one term
is odd and the other even.) In order that more than one pair of letters cancel, it
must be that α=s′′ and β =s′, so that m=n and
2 3
(s′α)m2−1 ·(βs′′)n−21 =s′′s′.
3 2 2 3
As before there is no further cancellationpossible if vs′s′′ is either the first term
3 2
in x3 or is preceded by a long term and vs′s′′ is either the last term in x2 or is
3 2
followed by a long term. In fact, only if s′3 =s′2 and vs′3s′2′ is followed in x2 by the
term s′ is there further cancellation. However, we are still left with a stray letter
2
s′′, so this product cannot occur.
2
However, if instead x3 ends with vs′3s′2′s′3 and x2 begins with vs′3s′2′, we obtain
the product s′3vs′2′s′3vs′3s′2′ = s′3 in the middle of x3x2. No further cancellation is
possible unless s′2 = s′3 and vs′3s′2′ is followed in x2 by the letter s′2. In this case,
it is easily seen that there are no more terms in either x or x , so that s′ = s′,
2 3 2 3
s′2′ = s′3′, and x2 = x3 = s′3vs′2′s′3. As before, w1 = w2, so w1s1w1−1 = s′1 and
w s w−1 = s′. Now, however, the vertex sˆ to which s is to be conjugated is
1 2 1 2 3 3
not s′3, but s′3′ = vs′3s′3′s′3vs−′31s′3′. This is seen by drawing the circuit C′ (compare
Lemma 4.3). But note
w1−1s′3′w1 =w3−1vs′3s′3′s′3s′3′s′3vs−′31s′3′w3 =w3s′3w3−1 =s3.
Therefore a single element again conjugates the vertices appropriately. Schemat-
ically, the two possibilities above can be summarized respectively as x = E,
3
x =E−1; and x =O, x =O−1 (as before, E =“even”,O =“odd”).
2 3 2
Similar arguments show that z = z = 1, that x cannot end with a short
2,3 1,2 3
term, and that x cannot begin with a short term. Thus the two products shown
2
above are the only valid possibilities when x =1.
1
Case 2. Now suppose that each x 6= 1, and that there is no cancellation of
i
commonshortterms. We givethe possible forms for x schematically (as was done
i
above) in the proposition below, leaving precise computations to the reader.
Proposition 4.1. Let x 6= 1 for i = 1,2,3. Up to renumbering, one of the
i
following holds:
x =E E , x =E−1E , x =E−1E−1; x =E E , x =E−1, x =E−1;
3 1 2 2 2 3 1 3 1 3 1 2 2 2 1 1
x =E O , x =O−1O , x =O−1E−1; x =E O , x =O−1, x =E−1;
3 1 1 2 1 2 1 2 1 3 1 1 2 1 1 1
x =E O , x =O−1E , x =E−1E−1 x =O E , x =E−1, x =O−1;
3 1 1 2 1 2 1 2 1 3 1 1 2 1 1 1
x =O O , x =O−1O , x =O−1O−1; x =O O , x =O−1, x =O−1.
3 1 2 2 2 3 1 3 1 3 1 2 2 2 1 1
Here, E represents either u or u β and O represents either v or v γ, for
j αβ αβ j γδ γδ
the appropriate choices of α, β, γ, and δ.
10 PATRICKBAHLS
Of course, one may not be able to choose freely whether E represents u or
j α,β
u β, and similarly for O . That is, the exact forms of the words E and O are
α,β j j j
clearly interdependent. (Cf. Section 4 of [3].)
Note that in multiplying any two distinct terms x and x , at most one long
i j
term may cancel. (That is, as we saw above, forms such as x = 1, x = O O ,
1 3 1 2
x = O−1O−1 cannot occur.) In fact, this will remain true even as we consider
2 2 1
arbitrarily long circuits.
Remark. There are a few cases which must be handled carefully; these cases
involvetheaffineEuclideanCoxetergroupswhosediagramsaretriangleswithedge
label multisets {2,3,6},{2,4,4}, or {3,3,3}. For instance, suppose that x =s′s′′
i i i
and s′ = s′′ for i = 1,2,3. Then x x x = 1, sˆ = s′ , and the edges of V′
i i+1 3 2 1 i i+1
which correspond to those in C do indeed form a circuit of length 3. However, it
can be shown (with the aid of Lemma 5.1) that no w ∈ W satisfies ws w−1 = sˆ.
i i
We claim that these forms of x lead to a fundamental contradiction. Consider
i
the parabolic subgroup w W w−1; because w s s−1 = s′, w s w−1 = s′′, and
1 C 1 1 1 1 1 1 3 1 3
w1s2w1−1 = s′3s′1s′3′s′1s′3, w1WCw1−1 ⊆ WC′. Now we apply the construction of xi
givenin Section 3 “inreverse”,proceeding from C′ to C. (Essentially, we compute
the ratios x′ = w−1 w .) We have proven our result for C (by conjugating back
i i−1 i
from the correspondingresult for C′) providedthe words x′ =w−1 w do not have
i i−1 i
formssimilartothoseofx . Inparticular,wemayassumethatx′ ∈W . Therefore,
i i C
sinces′3′ =s′2 andw1−1s′3w1 =w1−1w3s3w3−1w1 ∈WC,WC′ ⊆w1WCw1−1. Therefore
w1WCw1−1 =WC′. The set
{w s w−1 | i=1,2,3}={s′,s′′,s′s′s′′s′s′}
1 i 1 1 3 3 1 3 1 3
mustthereforegenerateWC′. However,it canbe shownthatthis setdoes notgen-
erateWC′ (thisispossiblebecausetheaffineEuclideangroupWC isnotcohopfian).
This gives a contradiction.
Therefore the case in which x =s′s′′ and s′′ =s′ for all i cannot occur. Any
i i i i i−1
similar case involving the affine Euclidean triangle groups can be outlawed in an
analogousfashion,andweareforcedtoconcludethattheformsforx ·x ·x given
1 2 3
above are exhaustive.
Before turning our attention to a proof of Theorem 2.3, we state the following
result concerning circuits of length 4.
Proposition 4.2. Let C be a circuit of length 4 and let x x x x = 1. Assume
4 3 2 1
furthermore that there is no cancellation of short terms between different words x
i
and x . Then, up to a renumbering of the vertices, there are 27 forms for the
j
product x ·x ·x ·x (entirely anologous to those given in Proposition 4.1). In
4 3 2 1
each case, every word x has at most two long terms, every long term of x cancels
i i
with a long term in either x or x , and no more than two long terms cancel
i+1 i−1
in any product x x .
i i−1
Onemaycheckthatineachofthecasesmentioned,theexactformsofthewords
O andE whichappeararedeterminedcompletely,andthatthe subdiagramofV′
i i
corresponding to these trivial products is a circuit of length 4 whose edges appear
inthe appropriateorder. This establishesTheorem2.2fork =4. (The formofthe
word which conjugates each edge [s s ] appropriately is easy to compute, given
i i+1
the forms of x and x .)
i+1 i