It is known after Stallings that a group can have 0, 1, 2 or infinitely many ends. Are there known results on the space of ends of a Coxeter group?
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1$\begingroup$ You can find some information on ends of Coxeter groups in this paper by Mihalik: sciencedirect.com/science/article/pii/0022404995001174 $\endgroup$– Nick GillCommented Sep 10, 2014 at 15:29
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5$\begingroup$ I think the theorem stating that a finitely generated group can only have 0, 1, 2 or infinitely many ends is rather due to Hopf. $\endgroup$– SeiriosCommented Sep 10, 2014 at 15:38
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$\begingroup$ @Seirios indeed it's due to Hopf and Freudenthal, both independently in the early 1940s. $\endgroup$– YCorCommented Sep 12, 2022 at 12:01
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1 Answer
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The following book has a wealth of material on this topic:
The geometry and topology of Coxeter groups by Michael Davis.
By way of example, here is one result from the book:
Theorem 8.7.1 Let $(W,S)$ be a Coxeter System:
$W$ is one-ended if and only $H_c^1(\Sigma)=0$.
$W$ has two ends if and only $H_c^1(\Sigma)\cong\mathbb{Z}$.
$W$ has infinitely many ends if and only if $H_c^1(\Sigma)$ has infinite rank.
Here $\Sigma$ is the cell complex associated to $(W,S)$. I have an e-copy of the book - email me if you want a copy.
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1$\begingroup$ I would not consider this as a satisfactory answer, since it changes a problem into another one. I would consider as a full answer one describing in terms of the combinatorial data defining the Coxeter group. For instance just describing when the Coxeter group is finite involves a quite lengthy list. Describing when it has 2 ends is just, in the connected case, the case of the infinite dihedral group. (...) $\endgroup$– YCorCommented Sep 10, 2014 at 17:42
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$\begingroup$ Here are 1 source of multi-ended Coxeter group: when there is a partition $S=A\cup C\cup B$ with $A,B$ non-empty, such that all edges between $A$ and $B$ are labeled $\infty$, and such that the subgroup generated by $C$ is finite. This is something checkable. Maybe there are other obvious sources of multi-ended Coxeter groups, and it would be natural to wonder if they are the only ones. $\endgroup$– YCorCommented Sep 10, 2014 at 17:46
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$\begingroup$ @YCor, of course your comment is entirely reasonable. There are plenty more results in the cited text that give more explicit information (see especially Thm. 8.7.3)... but I don't want to write them all out! $\endgroup$ Commented Sep 10, 2014 at 18:22
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$\begingroup$ Note: My comment above is addressed by Thomas Weigel in these 2018 slides ("An Ihara-type theorem for buildings"), for the "Trees, dynamics and locally compact groups" conference in Düsseldorf, Germany, 29.6.2018. These slides include a screenshot of this MO page (with comments), and include "In particular, Y. Cornulier’s question has an affirmative answer." I don't know if the author has attempted to publish this result. $\endgroup$– YCorCommented Sep 12, 2022 at 12:05