Question

In their treatise Groupes et algebres de Lie, Bourbaki (no doubt heavily influenced by Tits) devoted Chapter IV (1968) to the general theory of what they dubbed "Coxeter systems" $(W,S)$ along with "Tits systems" (BN-pairs). Here $S$ is an arbitrary set and $W$ a group generated by a subset $S$ consisting of elements of order 2, subject only to obvious relations involving pairs of generators. This is a very large class of groups, usually infinite, which includes finite reflection groups and others of interest in Lie theory. The axiomatic development in IV.1 doesn't require any restriction on the rank of the group: the cardinality of $S$.

On the other hand, there seem to be almost no significant examples in which the rank is infinite. As Bjorner and Brenti note in their book Combinatorics of Coxeter Groups, after defining Coxeter groups: "Most groups of interest will have finite rank." Typical examples given by them and others do include the group of permutations of the positive integers which leave all but finitely many fixed; this is a direct limit of finite symmetric groups (and embeds in the much larger "infinite symmetric group"). But although the general theory applies in all ranks, it's hard for me to think of anything really new one learns about infinite rank Coxeter groups using Coxeter theory. Maybe I haven't looked far enough, but it's natural to ask:

Are there significant results about Coxeter groups of infinite rank which aren't obtained just as easily without Coxeter theory?

Answer 1

This does not precisely answer your question, but whenever I see or write a result on Coxeter groups, I wonder whether it extends to infinite rank. In a number of cases, results on infinite rank formally follow from the study in finite rank, e.g. existence of a certain type of finitely generated subgroups, or the existence of some nice actions (e.g. on nonpositively curved cubical complexes), etc, but typically at the opposite, results asserting the existence of "many" quotients do not go to infinite rank which deserves a specific study (although this specific study ought to be based on techniques working in finite rank).

Example of a natural question: given a (say countable) Coxeter group, how many (according to its diagram) normal subgroups does it have: finitely many, infinitely countably many, or continuum? (it can't be otherwise by basic topology) The answer is known in f.g. case: it's at most countable iff all components are affine/finite (Gonciulea, Margulis-Vinberg) and finite iff the Coxeter group is finite (essentially obvious). In the infinitely generated case, when is it residually finite?

Sorry for self-advertising; here are a few papers where I evoke infinite rank Coxeter. In this paper with Stalder and Valette, we consider wreathed Coxeter groups, which are analogues of wreath products, based on a group action on a (usually infinite) Coxeter graph (Example 5.5). The terminology is from this paper with Bieri, Guyot, Strebel (see esp. Example 4.10), where the same groups are used as illustrations for completely different purposes. In this old unpublished note I address the simplicity of its $C^*$-algebra using a reduction to the f.g. case.