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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 '13 at 5:06
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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). –  user30379 Jan 31 '13 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 '13 at 10:43
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@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 '13 at 13:55
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@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! –  YCor Jan 31 '13 at 23:35

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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.

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@Yves: This is useful information (new to me), though I was wondering more naively about "significant" applications of Coxeter group theory in the infinite rank case relative to problems originating elsewhere. For me the prototypes of such problems occur in Lie theory rather than infinite group theory as such. But the study of Coxeter groups has gradually taken on its own identity, unrelated to Coxeter's geometric work or to Lie theory. –  Jim Humphreys Jan 31 '13 at 14:06
    
@Jim: yes: this was more an answer to the question in the title than this more specific question. Actually I don't have a very clear meaning what you want to include and discard in "Coxeter group theory", and I guess that if you want to solve the question I point out (which Coxeter groups have infinitely/uncountably many normal subgroup) then you may need a fine study using Coxeter theory. –  YCor Jan 31 '13 at 15:21
    
This paper arxiv.org/abs/0711.5011 by Dicks and Leary gives a characterization of Coxeter groups with finite virtual cohomological dimension (they include all f.g. ones, so the problem is really about infinitely generated ones). –  YCor Feb 1 '13 at 9:37

The following theorem was proved independently by Deodhar and by Dyer:

Let $(W, S)$ be a Coxeter group. Let $V$ be a subgroup of $W$ generated by reflections. Then there is a set $R$ of reflections in $V$ so that $(V, R)$ is a Coxeter group. $R$ can be characterized uniquely as follows: Let $T$ be the set of reflections in $W$ and, for $w \in W$, let $in(w)$ be the set of inversions of $w$. An element $t \in T$ lies in $R$ if and only if $in(t) \cap V = \{ t \}$.

Even when $S$ is finite, $R$ can be infinite! If I haven't messed up, one example is to take $W$ to be the Coxeter group on generators $p$, $q$, $r$, with no relations other than $p^2=q^2=r^2=e$. Let $V$ be the subgroup generated by $$\cdots, qpqprpqpq, qpqrqpq, qprpq, qrq, r, prp, pqrqp, pqprpqp, pqpqrqpqp, \cdots.$$ If I haven't messed up, $V$ is an infinite rank subgroup of $W$, with the set $R$ being the list of generators above.

Since these two papers have 86 citations in Mathscinet between them, I'd say this result is useful.

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Thanks for calling attention to this natural class of examples, which I had lost track of. In an infinite irreducible (but not affine) Coxeter group, there will be some reflection subgroups of infinite rank, due to the weird "geometry" of hyperplanes and roots coming from the Coxeter matrix. Your example of the universal group of rank 3 was analyzed in early work by Dyer, for example. These are certainly "significant" examples but I guess understood in detail only from properties of their finite Coxeter subgroups. –  Jim Humphreys Aug 22 '13 at 17:26
    
P.S. Concerning the citations, it may be instructive to follow them up on MathSciNet (though administrators only look at the numbers). Infinite rank probably doesn't matter so much, but the characterization of reflection subgroups as Coxeter groups is clearly important in the study of Coxeter groups for their own sake. I'm less certain about how the Dyer/Deodhar theorems interact with other subjects such as Kac-Moody theory –  Jim Humphreys Aug 22 '13 at 17:27

I don't think this will answer your direct question either, but it is still worth mentioning. The following fact was explained to me by a combination of Lusztig and Geordie Williamson, and it motivates why one should study (completions of) simply-laced Coxeter groups $(W',S')$ with $S'$ infinite.

There is a unique 2-sided cell $C$ in any Coxeter group $(W,S)$, consisting of all non-identity elements with a unique reduced expression. Any such element has a unique simple reflection in its right (resp. left) descent set. Within this cell, the left cells $C_s$ are parametrized by $s \in S$, and consist of all elements with a unique reduced expression and with right descent set $\{s\}$.

One can take the elements of $C_s$ and give them the structure of a graph, where $w$ is connected to $v$ if $w = tv$ for some $t \in S$. Each vertex can be labeled with the unique element of its left descent set. The resulting labeled graph does not depend on the choice of $s$.

One can view this graph as encoding a simply-laced Coxeter group $(W',S')$. Suppose that $S'$ is finite. For each $t \in S$, the reflections labeled by $t$ all mutually commute, and their product is an involution in $W'$. Together, these involutions generate a subgroup inside $W'$ which is isomorphic to the original Coxeter group $W$. In this way, any finite Coxeter group is canonically embedded inside a simply-laced one. Any Coxeter element of $W$ is sent to a Coxeter element of $W'$.

For example, this operation will produce the embedding of $H_4$ inside $E_8$. It will produce the embedding of a finite dihedral group $I_2(m)$ into $A_{m-1}$, sending each simple reflection in $I_2(m)$ to a product of every other simple reflection in $A_{m-1}$.

However, it may happen that $S'$ is infinite, and that the collection of vertices labeled by $t \in S$ is also infinite. In this case, one does not have an embedding of $W$ into $W'$, because each simple reflection would have to go to an infinite product. Though I have not seen it defined and don't know the literature, I imagine there is an embedding of $W$ into some suitable completion of $W'$.

For instance, this gives the embedding of $I_2(\infty)$ into $A_{\infty}$, sending each simple reflection in $I_2(\infty)$ to the product of every other simple reflection in $A_{\infty}$.

Anyway, the upshot of all this is that one could potentially study arbitrary Coxeter groups (with $S$ finite) using only (completions of) simply-laced ones (with $S'$ possibly infinite).

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Thanks for these interesting observations. (Though I'm not quite ready to sign on to the program in your last sentence.) –  Jim Humphreys Aug 22 '13 at 17:24

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