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Discrete group extensions of countable Markov shifts and pressure gap
The main outcome of this note is to extend the work of Stadlbauer (\textit{An extension of Kesten's criterion for amenability to topological Markov chains}, 2013), Coulon--Dal'Bo--Sambusetti (\textit{Growth gap in hyperbolic groups and amenability}, 2018), and Dougall (\textit{Critical exponents of
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