Symmetries of Hyperbolic 4-Manifolds. (25th July 2015) - Kolpakov, Alexander; Slavich, Leone
Abstract : In this paper, for each finite group $G$, we construct the first explicit examples of non-compact complete finite-volume arithmetic hyperbolic 4-manifolds $M$ such that ${\rm Isom}\, M \cong G$, or ${\rm Isom}^{+}\, M \cong G$ . In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic 4-space, on the one hand, and the combinatorics of simplicial complexes, on the other hand. This allows us to obtain a universal upper bound on the minimal volume of a hyperbolic 4-manifold realizing a given finite group $G$ as its isometry group in terms of the order of the group. We also obtain asymptotic bounds for the growth rate, with respect to volume, of the number of hyperbolic 4-manifolds having a finite group $G$ as their isometry group.