1. Introduction

A Coxeter group is determined by its diagram It is known that in certain cases, determines as well (see, e.g., [1, 2]). This is the case for right-angled Coxeter groups [3, 4], where the only relations are for all generators and for some pairs of generators and For right-angled Coxeter groups, it is convenient to consider the Coxeter diagram (rather than the classical Coxeter graph): the presence of an edge with endpoints and means that and commute in

The properties of a right-angled Coxeter group depend almost exclusively on the combinatorics of the diagram This is especially evident in the study of For example, the groupoid consisting of the vertex sets of complete subgraphs of plays an important role in [5] where Tits exhibits a split exact sequence Tits also defines to have propriété I if the complementary graph has no triangles. He goes on to show that if has propriété I, then is isomorphic to

In the present article, we focus on the set whose members are the vertex sets of maximal complete subgraphs. We say that has condition C if there exist and a collection such that (C1) for (C2) for each the cardinality of the set is odd. When has condition C, the subgraph spanned by has propriété I. Thus, our condition C is a “local version” of Tits' propriété I.

Motivated by the results of Tits regarding sequence (1.1), we consider the sequenceA splitting of this sequence implies that all extensions of the formare trivial (cf. [6]). Clearly (1.2) does not always split. For example, is a right-angled Coxeter group andis a nontrivial extension. On the other hand, if has no center, then finding nontrivial extensions of is surprisingly difficult. Whether (1.2) splits for all with trivial center is currently an open question.

The group has been studied extensively in [5, 7] and is called the group of symmetric automorphisms of We approach the problem of whether (1.2) splits by considering the following.

(a) Does the sequencesplit?(b) Are the splittings of (1.1) and (1.5) compatible? A positive answer to both (a) and (b) implies that (1.2) is a split extension. We show in Theorem 5.4 that if has condition C, then (1.5) splits. To obtain a splitting of (1.2), we show that the action of on is compatible if is not “too symmetrical.” More precisely, our main result is the following.

Theorem 1.1. If has condition C and each leaves all vertices of invariant, then (1.2) is a split exact sequence.

It was recently shown [8] that (1.5) always splits. However, this result does not lead to a generalization of Theorem 1.1 as the splitting given there is not, in general, compatible with (1.1).

2. Right-Angled Coxeter Groups

Coxeter groups are typically defined by presentations, and there are various conventions for representing such presentations diagramatically. In this section, we review some definitions and important properties, focusing exclusively on the right-angled case. See [9] or [10] for a comprehensive treatment.

If is any set, let denote the set of subsets of with cardinality 2. Definition 2.1. Given a finite set and let denote the undirected graph with vertex set and edge set (note that does not have loops or parallel edges). As such graphs are often used to represent right-angled Coxeter groups, is called a Coxeter diagram.Definition 2.2. Given set The graph is the subgraph of spanned by A complete subgraph is maximal if it is not properly contained in any complete subgraph of .Definition 2.3. The presentationis the Coxeter presentation defined by Definition 2.4. A group is a right-angled Coxeter group if it has a presentation defined by some Coxeter diagram In this case, one writes   and calls the right-angled Coxeter group defined by The pair is a right-angled Coxeter system.Remark 2.5. The Coxeter diagram is not the same as the traditional Dynkin diagram. Indeed, as graphs, the Coxeter and Dynkin diagrams are complementary.

Clearly, each Coxeter diagram defines a unique right-angled Coxeter group. On the other hand, to recover the diagram from a group one must first choose a particular Coxeter presentation. It is natural to wonder whether nonisomorphic diagrams might define isomorphic groups. The relationship between right-angled Coxeter groups and their diagrams is clarified by the following result.

Theorem 2.6 (Radcliffe [3]). If and are right-angled Coxeter systems for then there is an automorphism such that

A similar result was also obtained by Castella in [4]. Definition 2.7. A subgroup of generated by a subset of is called a special subgroup. If it is customary to write for the subgroup generated by Finite special subgroups are called spherical subgroups. A subgroup of is parabolic if it is conjugate to a special subgroup.

We conclude this section with statements of some of the remarkable properties enjoyed by special subgroups. For proofs, see [9] or [10]. Theorem 2.8. If then Theorem 2.9. If then is the right-angled Coxeter group defined by Corollary 2.10. The following are equivalent:
(a) is spherical;(b) is an elementary abelian 2-group of rank (c) is complete.

3. Automorphisms of Right-angled Coxeter Groups

For the remainder of this article, let be the right-angled Coxeter group defined by the connected Coxeter diagram It is easily verified that, under the operation of symmetric difference, the setis a commutative groupoid with identity

In [5], Tits uses the group of automorphisms of to exhibit as a semidirect product. We sketch the construction. Let It is well known that every maximal finite subgroup of is parabolic (see, e.g., [11, Lemma 3.2.1]). Consequently, every finite subgroup of is conjugate into a spherical subgroup. It follows that, for each there is a unique minimal such that is conjugate into The mapgiven by is an epimorphism. Definition 3.1. Let be the kernel of Elements of are called symmetric automorphisms of

Given consider defined byfor all A main result of [5] is the following theorem. Theorem 3.2. The mapping given by is a splitting of the sequence

Let be the standard path metric on that assigns length one to each edge. For each define The subgraph spanned by the elements of gives rise to certain generators of as follows. If is the vertex set of a connected component of then the map given byextends to a unique involution The following is easily deduced from [7]. Theorem 3.3. For each let be the vertex sets of the components of Then is generated by the set Remark 3.4. In [7], Mühlherr gives a complete presentation for based on a slightly different set of generators.

4. Symmetric Automorphisms

The subsets of that generate maximal spherical subgroups of play a key role in the subsequent development. As such, we defineNote that is in one-to-one correspondence with the family of maximal complete subgraphs of . The global behavior of a symmetric automorphism is governed by the following observation: if and then there exists an element such that for all Remark 4.1. When is its own centralizer. Thus, the element above is determined up to right multiplication by any member of (i.e., up to choice of representative for the coset ).

If and then for all Consequently, lies in the centralizer of Definition 4.2. With as above, let A representative of the double coset is called a -transition from to Remark 4.3. If have pairwise nonempty intersection, then and 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 element For much of what follows, may be chosen arbitrarily; in Section 5 we show that a preferred choice may exist.

Given let be the inner automorphism of given by for all Restricting our attention to we define In other words, is the pointwise stabilizer of under the action of on W. Observe that, since the subgraph spanned by is complete, is Abelian.

Clearly, is a subgroup of Let be the restriction to of the natural map and choose a class with representative If is the restriction of to then for all Since and it follows that is onto. We have established the following. Theorem 4.4. The sequence is a central extension.

In Section 5, we construct a retraction of the mapping We now describe the key ingredient used in this construction. For each letIt follows at once that is the centralizer of in (cf. [5, page 350]). The functiongiven byextends uniquely to a retraction which we also denote by

5. Splittings

As in Section 4, we assume that is a fixed element of Definition 5.1. One says that satisfies condition C if there exist elements such that the following conditions hold.
(C1) for each (C2) For each the cardinality of the set is odd. If contains a maximal complete subgraph whose vertex set satisfies condition C, then one says that has condition C. When has condition C, then for each above let denote the maximal complete subgraph of spanned by Note that, since is connected, when our hypotheses imply that is complete. In this case, is Abelian and the results below are trivial. Thus, we may assume that is a positive integer. Example 5.2. We illustrate condition C with some examples.
(a) If contains a maximal complete subgraph with an even number of vertices each of which meets exactly one other complete subgraph of then has condition C. For example, any Coxeter diagram that contains Figure 1(a) as a subgraph (where , , and are maximal complete subgraphs of ) has condition C.(b) Suppose is odd and has maximal complete subgraphs If the vertex set of is and the vertex set of contains every element of but then has condition C. For example, any Coxeter diagram that contains Figure 1(b) as a subgraph (where , and are maximal complete subgraphs of ) has condition C.

Figure 1

Remark 5.3. In [5], Tits says that a right-angled Coxeter group with Coxeter diagram has “propriété I” if the complementary graph has no triangles. In this case, the inclusion is an isomorphism. If has condition C, then the union defines a right-angled Coxeter group that has propriété I. Thus, condition C is in some sense a “local” version of Tits' propriété I.

Assume satisfies condition C and let In this case, we can choose Then, for each with the transition is an element of the coset It must be emphasized that no left multiplication by elements of is permitted. Theorem 5.4. If has condition C, then the sequence splits.

Proof. Let satisfying (C1) and (C2) of Definition 5.1. For each define an inner automorphism where(recall that is the inner automorphism that conjugates each element of by ). By definition, lies in . Consequently, the terms in the product (5.2) commute and so the mappingis well defined. To see that is a homomorphism, choose Then, for each we haveSince lies in the centralizer of any reduced expression for is of the form where each Observe thatand so It follows from (5.4) thatNow, using the fact that the image of each lies in the Abelian subgroup we have that
To see that is a retraction, let and let be a number of 's that do not contain If is the inner automorphism of that conjugates by then for every From (5.2) we have that conjugates every element bySince is odd, and so
Since is a retraction and the sequence (4.3) is central, the mappingdefined byis an isomorphism. Consequently, is an isomorphism from onto A section is given by the composition where is the inclusion

As noted in the introduction, to obtain a splitting of sequence (1.2), must satisfy an “asymmetry” condition. This is obtained by imposing a restriction on the action of and motivates the following definition. Definition 5.5. Let and One says that fixes if for every

Note that if fixes then for everyTheorem 5.6. If has condition C and each fixes then splits.

Proof. Let Since each fixes it follows that (as defined in (3.3) above) lies in and the mapping is a splitting of the sequence(where is the restriction of the mapping given in (3.2) above). As in the proof of Theorem 4.4, the projection is onto and its kernel isBut and so Thus, we have an extensionwhich is easily seen to be central.
If and then and, because is Abelian,Since (5.12) splits, every element has a unique expression where and Consequently, we may defineby for all If is written as , then, by (5.15),Thus, is a homomorphism and is easily verified to be a retraction. The proof is completed in a manner similar to the conclusion of the proof of Theorem 5.4, replacing with and so on.

As an immediate consequence we obtain the following corollary (cf. [6]). Corollary 5.7. If is a right-angled Coxeter group satisfying the hypotheses of Theorem 5.6, then every extension splits.Example 5.8. The graph in Figure 2 clearly satisfies the hypothesis of Theorem 5.6. On the other hand, Theorem 5.6 does not apply to either of the graphs given in Figure 1.

Figure 2

Remark 5.9. As noted in the introduction, it is currently unknown whether, for with trivial center, the sequencealways splits. However, it was recently shown in [8] that sequence (1.5) always splits, but the splitting found there is not, in general, compatible with (1.1). In particular, one cannot obtain generalizations of Theorem 5.6 and Corollary 5.7 from [8].