Entropy and drift for word metric on relatively hyperbolic groups

Matthieu Dussaule (1) , Ilya Gekhtman

1 LMJL - Laboratoire de Mathématiques Jean Leray

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 Blachère, Haïssinsky 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.

Domaines: Théorie des groupes [math.GR]; Systèmes dynamiques [math.DS]; Probabilités [math.PR]

Matthieu Dussaule, Ilya Gekhtman. Entropy and drift for word metric on relatively hyperbolic groups. 2019. ⟨hal-01935131v3⟩