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## Is there any need to study Coxeter systems (W,S) with S infinite?

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?

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Infinite rank (even of cardinality continuum) Coxeter groups appear naturally in the theory of nondiscrete affine buildings, e.g. asymptotic cones if symmetric spaces. However, the Coxeter theory in this setting is rather trivial. – Misha Jan 31 at 5:06
I once asked a member of the 2nd generation of Bourbaki why they decided to develop the theory without assuming $S$ to be finite. The answer I got was that the proofs work for general $S$, so they decided to write it that way (and that there were no specific examples of interest with infinite $S$ that they had in mind). – pranavk Jan 31 at 5:27
Isn't an infinite rank Coxeter group just the direct limit of its finite rank parabolics and so the situation is always like the infinite symmetric group? – Benjamin Steinberg Jan 31 at 10:43
@pranavk: This confirms my suspicions. (I never thought to ask any of the insiders decades ago.) In mathematics one often looks for the natural generality in which to state things, but that can be a hard sell to students who need to get on to their own work. – Jim Humphreys Jan 31 at 13:55
@Benjamin: I don't agree: the infinite finitary symmetric group seems to be quite exceptional among infinitely generated Coxeter groups, since it has the property that all finite rank parabolics are finite. There are only 4 irreducible infinite rank Coxeter systems with this property namely: the two $A_\infty$ (one-sided infinite or two-sided infinite), $B_\infty$, and $D_\infty$. Any other irreducible infinite rank Coxeter group contains a free subgroup on 2 generators! – Yves Cornulier Jan 31 at 23:35

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.