The Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups
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Publication:6294629
DOI10.4171/LEM/66-3/4-3zbMath1511.20156arXiv1711.11307WikidataQ115481576 ScholiaQ115481576MaRDI QIDQ6294629
Victor Gerasimov, Matthieu Dussaule, Leonid Potyagailo, Ilya Gekhtman
Publication date: 30 November 2017
Abstract: 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.
Geometric group theory (20F65) Topological methods in group theory (57M07) Martin boundary theory (31C35) Hyperbolic groups and nonpositively curved groups (20F67) General properties and structure of locally compact groups (22D05) Probability measures on groups or semigroups, Fourier transforms, factorization (60B15)
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