An extension of Kesten's criterion for amenability to topological Markov chains

TL;DR

This paper extends Kesten's criterion for amenability from symmetric random walks on groups to group extensions of topological Markov chains, establishing conditions under which amenability relates to Gurevic-pressures and providing an application to hyperbolic manifolds.

Contribution

It generalizes Kesten's criterion to topological Markov chains with new conditions linking amenability and pressure, including an application to hyperbolic geometry.

Findings

Amenability implies equality of Gurevic-pressures under mild conditions.

Hölder continuity and big images condition ensure the converse.

Application to periodic hyperbolic manifolds demonstrates practical relevance.

Abstract

The main results of this note extend a theorem of Kesten for symmetric random walks on discrete groups to group extensions of topological Markov chains. In contrast to the result in probability theory, there is a notable asymmetry in the assumptions on the base. That is, it turns out that, under very mild assumptions on the continuity and symmetry of the associated potential, amenability of the group implies that the Gurevic-pressures of the extension and the base coincide whereas the converse holds true if the potential is H\"older continuous and the topological Markov chain has big images and preimages. Finally, an application to periodic hyperbolic manifolds is given.