Entropy and drift for word metric on relatively hyperbolic groups

DOI10.4171/GGD/588arXiv1811.10849WikidataQ115481607 ScholiaQ115481607MaRDI QIDQ6310273

Publication date: 27 November 2018

Abstract: We are interested in the Guivarc'h inequality for admissible random walks on finitely generated relatively hyperbolic groups, endowed with a word metric. We show that for random walks with finite super-exponential moment, if this inequality is an equality, then the Green distance is roughly similar to the word distance, generalizing results of Blach{`e}re, Ha{"i}ssinsky and Mathieu for hyperbolic groups [4]. Our main application is for relatively hyperbolic groups with respect to virtually abelian subgroups of rank at least 2. We show that for such groups, the Guivarc'h inequality with respect to a word distance and a finitely supported random walk is always strict.

Mathematics Subject Classification ID

Geometric group theory (20F65) Asymptotic properties of groups (20F69) Hyperbolic groups and nonpositively curved groups (20F67) Boundary theory for Markov processes (60J50) Random walks on graphs (05C81)

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