Hyperbolic Coxeter groups of large dimension

Abstract

We construct examples of Gromov hyperbolic Coxeter groups of arbitrarily large dimension. We also extend Vinberg's theorem to show that if a Gromov hyperbolic Coxeter group is a virtual Poincaré duality group of dimension $n$, then $n\le 61$. Coxeter groups acting on their associated complexes have been extremely useful source of examples and insight into nonpositively curved spaces over last several years. Negatively curved (or Gromov hyperbolic) Coxeter groups were much more elusive. In particular, their existence in high dimensions was in doubt. In 1987 Gabor Moussong [M] conjectured that there is a universal bound on the virtual cohomological dimension of any Gromov hyperbolic Coxeter group. This question was also raised by Misha Gromov [G] (who thought that perhaps any construction of high dimensional negatively curved spaces requires nontrivial number theory in the guise of arithmetic groups in an essential way), and by Mladen Bestvina [B2]. In the present paper we show that high dimensional Gromov hyperbolic Coxeter groups do exist, and we construct them by geometric or group theoretic but not arithmetic means.