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
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?