Drift and Matrix coefficients for discrete group extensions of countable Markov shifts

TL;DR

This paper extends Kesten's criterion for amenability to discrete group extensions of countable Markov shifts, using matrix coefficients and twisted measures to analyze spectral properties and decay rates.

Contribution

It generalizes amenability criteria to countable Markov shifts with group extensions and develops a representation-theoretic framework involving twisted measures.

Findings

Recovers criteria for amenability via matrix coefficient decay.

Recasts Gurevič pressure drops in terms of decay of twisted measures.

Locates previous results within a new representation-theoretic framework.

Abstract

There has been much interest in generalizing Kesten's criterion for amenability in terms of a random walk to other contexts, such as determining amenability of a deck covering group by the bottom of the spectrum of the Laplacian or entropy of the geodesic flow. One outcome of this work is to generalise the results to so-called discrete group extensions of countable Markov shifts that satisfy a strong positive recurrence hypothesis. The other outcome is to further develop the language of unitary representation theory in this problem, and to bring some of the machinery developed by Coulon-Dougall-Schapira-Tapie [Twisted Patterson-Sullivan measures and applications to amenability and coverings, arXiv:1809.10881, 2018] to the countable Markov shift setting. In particular we recast the problem of determining a drop in Gurevi\v{c} pressure in terms of eventual almost sure decay for matrix coefficients, and explain that a so-called twisted measure "finds points with the worst decay." We are also able locate the results of Dougall-Sharp [Anosov flows, growth rates on covers and group extensions of subshifts, {\em Inventiones Mathematicae}, 223, 445-483, 2021] within this framework.