Entropy and drift for word metrics on relatively hyperbolic groups

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 applications are for relatively hyperbolic groups with some virtually abelian parabolic subgroup of rank at least 2, relatively hyperbolic groups with spherical Bowditch boundary, and free products with at least one virtually nilpotent factor. 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.