ENTROPY AND DRIFT FOR WORD METRICS 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 hy-perbolic 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 METRICS ON RELATIVELY HYPERBOLIC GROUPS. 2018. ⟨hal-01935131v1⟩