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

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.