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This question came up in my abstract algebra class today, and cursory research did not find a satisfactory answer.

Which Coxeter groups are hyperbolic groups (in the sense of Gromov)? I know that some classes of them are, but are there any counterexamples?

mathperson314
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    @WillJagy: Humphreys uses different definition of hyperbolicity: OP had Gromov-hyperbolicity in mind. Gromov-hyperbolic Coxeter groups exist in all dimensions. – Moishe Kohan Dec 05 '24 at 02:33
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    A finitely generated Coxeter group is hyperbolic if and only if it does not contain $\mathbb Z^2$ (this is Moussong's theorem). – Moishe Kohan Dec 05 '24 at 02:33
  • @MoisheKohan thanks. I guess I'll leave the screen capture for now; perhaps the OP will be more specific – Will Jagy Dec 05 '24 at 02:59

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G. Moussong in his PhD thesis proved the following characterization of Gromov-hyperbolic Coxeter groups:

Theorem. Suppose that $W$ is a finitely generated Coxeter group with labelled Coxeter graph $G$. Then the following are equivalent:

  1. $W$ is Gromov-hyperbolic.

  2. $W$ does not contains $\mathbb Z^2$.

  3. The graph $G$ does not contain subgraphs corresponding to:

(i) Euclidean Coxeter groups of dimension $\ge 2$.

(ii) Products of infinite Coxeter groups $W_1\times W_2$.

You can find details in

Davis, Michael W., The geometry and topology of Coxeter groups., London Mathematical Society Monographs Series 32. Princeton, NJ: Princeton University Press (ISBN 978-0-691-13138-2/hbk). xiv, 584 p. (2008). ZBL1142.20020.

Gromov-hyperbolic Coxeter groups exist in all dimensions:

Januszkiewicz, Tadeusz; Świątkowski, Jacek, Hyperbolic Coxeter groups of large dimension., Comment. Math. Helv. 78, No. 3, 555-583 (2003). ZBL1068.20043.

For a simpler construction see

Osajda, Damian, A construction of hyperbolic Coxeter groups., Comment. Math. Helv. 88, No. 2, 353-367 (2013). ZBL1282.20049.

Moishe Kohan
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