Recurrence and pressure for group extensions

TL;DR

This paper studies the thermodynamic formalism for recurrent potentials in group extensions of countable Markov shifts, revealing conditions for recurrence, amenability, and pressure equivalences, with applications to Kleinian groups and group presentations.

Contribution

It characterizes recurrent potentials depending only on the base space and links recurrence to group amenability, providing new insights into pressure and group dynamics.

Findings

Recurrent potentials depend only on the base space under certain conditions.

Groups are necessarily amenable if the potential depends only on the base.

Conditions are provided for the equality of base and skew product pressure.

Abstract

We investigate the thermodynamic formalism for recurrent potentials on group extensions of countable Markov shifts. Our main result characterises recurrent potentials depending only on the base space, in terms of the existence of a conservative product measure and a homomorphism from the group into the multiplicative group of real numbers. We deduce that, for a recurrent potential depending only on the base space, the group is necessarily amenable. Moreover, we give equivalent conditions for the base pressure and the skew product pressure to coincide. Finally, we apply our results to analyse the Poincar\'e series of Kleinian groups and the cogrowth of group presentations.