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- W147370547 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 im- plies that the Gureviy of the extension and the base coincide whereas the converse holds true if the potential is Holder continuous and the topological Markov chain has big images and preimages. Finally, an application to periodic hyperbolic manifolds is given. MSC 2000. 37A50, 37C30, 20F69 1 Introduction and statement of main results The motivation for the analysis of the change of pressure under group extensions stems from the attempt to relate two classical results from probability theory and geometry on the amenability of discrete groups. The probabilistic result was obtained by Kesten in (11) and characterises amenability in terms of the spectral radius of the Markov operator associated to a symmetric random walk, that is a group G is amenable if and only if the spectral radius of the operator acting on l 2 (G) is equal to 1. The following counterpart in geometry was discovered by Brooks ((3)) using a completely different method. Assume that G is a Kleinian group acting on hyperbolic space H n+1 with exponent of convergence δ(G) bigger than n/2 and that N ⊳G is a normal subgroup. Then the bottoms of the spectra of the Laplacians on H/G and H/N are equal if and only if G/N is amenable. Or" @default.
- W147370547 created "2016-06-24" @default.
- W147370547 creator A5059614045 @default.
- W147370547 date "2011-10-13" @default.
- W147370547 modified "2023-09-27" @default.
- W147370547 title "An extension of a theorem of Kesten to topological Markov chains" @default.
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