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

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