Symmetries of hyperbolic 4-manifolds

TL;DR

This paper constructs explicit examples of hyperbolic 4-manifolds with prescribed finite symmetry groups using Coxeter polytopes and simplicial complex combinatorics, advancing the understanding of symmetries in hyperbolic geometry.

Contribution

It provides explicit constructions of hyperbolic 4-manifolds with any given finite symmetry group, combining geometric and combinatorial methods.

Findings

Explicit hyperbolic 4-manifolds with prescribed symmetry groups

Use of Coxeter polytopes in construction

Application of simplicial complex combinatorics

Abstract

In this paper, for each finite group $G$, we construct explicitly a non-compact complete finite-volume arithmetic hyperbolic $4$-manifold $M$ such that $\mathrm{Isom}\,M \cong G$, or $\mathrm{Isom}^{+}\,M \cong G$. In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic $4$-space, on one hand, and the combinatorics of simplicial complexes, on the other.