An extension of Kesten's criterion for amenability to topological Markov chains

Abstract

The main results of this note extend a theorem of Kesten for symmetric random walks on discrete groups to group extensions of topological Markov chains. In contrast to the result in probability theory, there is a notable asymmetry in the assumptions on the base. That is, it turns out that, under very mild assumptions on the continuity and symmetry of the associated potential, amenability of the group implies that the Gurevic-pressures of the extension and the base coincide whereas the converse holds true if the potential is Hölder continuous and the topological Markov chain has big images and preimages. Finally, an application to periodic hyperbolic manifolds is given.

New proof of Lemma 5.3 due to the gap in the first version of the article

References in corpus (1)

• Fractal Models for Normal Subgroups of Schottky Groups

Cited by in corpus (14)

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• Fractal Models for Normal Subgroups of Schottky Groups
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• Diophantine properties of groups of toral automorphisms
• A Lower Bound for the Exponent of Convergence of Normal Subgroups of Kleinian Groups
• Thermodynamic formalism for transient dynamics on the real line
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• The Martin boundary of an extension by a hyperbolic group
• Growth gap in hyperbolic groups and amenability
• Group Extended Markov Systems, Amenability, and the Perron-Frobenius Operator
• Conformal Fractals for Normal Subgroups of Free Groups
• Critical exponents of normal subgroups, the spectrum of group extended transfer operators, and Kazhdan distance
• Discrete group extensions of countable Markov shifts and pressure gap