Kesten's theorem for uniformly recurrent subgroups

2018/01/27 by Fraczyk, Mikolaj · 2 citations
#05C81 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1801.09132

Abstract

We prove an inequality on the difference between the spectral radius of the Cayley graph of a group G and the spectral radius of the Schreier graph H\backslash G for any subgroup H. As an application we extend Kesten's theorem on spectral radii to uniformly recurrent subgroups and give a short proof that the result of Lyons and Peres on cycle density in Ramanujan graphs holds on average. More precisely, we show that if \mathcal G is an infinite deterministic Ramanujan graph, then the time spent in short cycles by a random walk of length n is o(n).

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• Elementwise conservative actions and new constructions of boomerang subgroups
• Critical exponents of boomerang subgroups in the free group

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