Properties of sets of isometries of Gromov hyperbolic spaces

TL;DR

This paper establishes a new inequality for isometries in Gromov hyperbolic spaces, extending spectral radius concepts and demonstrating continuity and analogues of classical theorems without requiring properness or geodesicity.

Contribution

It introduces a hyperbolic joint stable length and proves an inequality that generalizes spectral radius properties to isometries of Gromov hyperbolic spaces.

Findings

Proves an inequality involving the joint stable length of isometries.

Shows the continuity of the joint stable length.

Provides an analogue of the Berger-Wang theorem in this setting.

Abstract

We prove an inequality concerning isometries of a Gromov hyperbolic metric space, which does not require the space to be proper or geodesic. It involves the joint stable length, a hyperbolic version of the joint spectral radius, and shows that sets of isometries behave like sets of $2 \times 2$ real matrices. Among the consequences of the inequality, we obtain the continuity of the joint stable length and an analogue of Berger-Wang theorem.