Hyperbolic Coxeter groups of large dimension.
For any \(n\) the authors construct a Gromov hyperbolic Coxeter group \(W\) with virtual cohomological dimension \(\text{vcd}(W)=n\). The question of existence such groups with large vcd was raised by \textit{G. Moussong} [Hyperbolic Coxeter groups, Ph.D. thesis, Ohio State Univ. (1987)], \textit{M. Gromov} [Geometric group theory. Volume 2: Asymptotic invariants of infinite groups. (1993; Zbl 0841.20039)] and Bestvina (on his problem list). The result is surprising in light of the conjecture of Moussong that there should be a universal bound on the vcd of any Gromov hyperbolic Coxeter group and the belief expressed by Gromov that any construction of high dimensional negatively curved spaces would involve, in an essential way, nontrivial number theory in the guise of arithmetic groups. The authors give an explicit construction of these groups using complexes of groups techniques from \textit{M. R. Bridson} and \textit{A. Haefliger} [Metric spaces of non-positive curvature. (Grundlehren der Math. Wiss. 319, Springer) (1999; Zbl 0988.53001)]. The authors contrast their groups of large dimension with fundamental groups of closed aspherical manifolds by extending a result of \textit{Eh. B. Vinberg} [Usp. Mat. Nauk 40, No. 1(241), 29-66 (1985; Zbl 0579.51015)]. They show that if \((W,S)\) is a virtual Poincaré duality Coxeter group of dimension \(n\) that is Gromov hyperbolic, then \(n\leq 61\). Moreover, in the case that \((W,S)\) is right angled, \(n\leq 4\) and this upper bound is sharp. As a result they conclude that if \(M\) is a manifold with dimension greater than \(4\), then \(M\) admits no negatively curved piecewise hyperbolic cubical metric.
scientific article; zbMATH DE number 2021085