Asymptotic properties of group actions and topological versions of Kesten’s theorem

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2023-07-14

Authors

Chaudkhari, Maksym

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Abstract

We study geometric and probabilistic properties of group actions and combinatorial extensions of Kesten’s theorem to the general case of amenable topological groups. In particular, we provide a characterization of confined subgroups of Thompson’s group F in terms of its action on the unit interval. Furthermore, we obtain the results connecting the uniform Liouville property of the group actions on the orbits of a countable Borel equivalence relation R with its amenability, if the acting group is dense in the full group of R. We discuss possible extensions of Kesten’s theorem to the general setting of amenable Hausdorff topological groups and prove a version of this theorem for the groups with SIN property. Moreover, we also describe possible applications of the topological version of Kesten’s theorem to the concentration inequalities for inverted orbits and extensive amenability of group actions.

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