Hyperbolic Coxeter groups of large dimension

We construct examples of Gromov hyperbolicCoxeter groups of arbitrarily large dimension.We also extend Vinbergs theorem to show that if a Gromovhyperbolic Coxeter group is a virtual Poincaré duality groupof dimension n, then n 61.Coxeter groups acting on their associated complexes have been extremelyuseful source of examples and insight into nonpositively curved spacesover last several years. Negatively curved (or Gromov hyperbolic)Coxeter groups were much more elusive. In particular their existence inhigh dimensions was in doubt.In 1987 Gabor Moussong [M] conjectured that there is a universal bound onthe virtual cohomological dimension of any Gromov hyperbolic Coxeter group.This question was also raised by Misha Gromov [G] (who thought that perhapsany construction of high dimensional negatively curved spaces requiresnontrivial number theory in the guise of arithmetic groups in an essentialway), and by Mladen Bestvina [B2].In the present paper we show that high dimensional Gromov hyperbolic Coxetergroups do exist, and we construct them by geometric or group theoretic butnot arithmetic means.