The Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups

Matthieu Dussaule (1) , Ilya Gekhtman , Victor Gerasimov , Leonid Potyagailo (2)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 LPP - Laboratoire Paul Painlevé - UMR 8524

Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green's function of the corresponding random walk. We give a complete topological characterization of the Martin boundary of finitely supported random walks on relatively hyperbolic groups with virtually abelian parabolic subgroups. In particular, in the case of nonuniform lattices in the real hyperbolic space H n , we show that the Martin boundary coincides with the CAT (0) boundary of the truncated space, and thus when n = 3, is homeomorphic to the Sierpinski carpet.

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

Matthieu Dussaule, Ilya Gekhtman, Victor Gerasimov, Leonid Potyagailo. The Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups. 2017. ⟨hal-01652248v1⟩