Finiteness Theorems for Gromov-Hyperbolic Spaces and Groups

TL;DR

This paper establishes finiteness results for torsion-free groups acting on hyperbolic spaces with bounded entropy and compact quotients, leading to topological and homotopical finiteness theorems for related spaces.

Contribution

It proves finiteness theorems for certain hyperbolic groups and spaces based on entropy and hyperbolicity bounds, providing explicit estimates for their number.

Findings

Finite set of torsion-free hyperbolic groups with bounded entropy and compact quotient.

Finiteness of non-cyclic torsion-free hyperbolic marked groups with bounded entropy.

Homotopical and topological finiteness results for related metric spaces and manifolds.

Abstract

In this article we prove that the set of torsion-free groups acting by isometries on a hyperbolic metric space whose entropy is bounded above and with a compact quotient is finite. The number of such groups can be estimated in terms of the hyperbolicity constant and of an upper bound of the entropy of the space and of an upper bound of the diameter of its quotient. As a consequence we show that the set of non cyclic torsion-free $\delta$-hyperbolic marked groups whose entropy is bounded above by a number $H$ is finite with cardinality depending on $\delta$ and $H$ alone. From these results, we draw homotopical and topological finiteness theorems for compact metric spaces and manifolds.