Symmetries of Hyperbolic 4-Manifolds. (25th July 2015)

• Record Type: Journal Article
• Authors: Kolpakov, Alexander; Slavich, Leone
• Is Part Of: International mathematics research notices. Volume 2016:Number 9(2016)
• Page Start: 2677
• Page End: 2716
• Publication Date: 2015-07-25
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnv210
• Languages: English
• ISSNs: 1073-7928

Abstract

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.