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The Martin Boundary of Relatively Hyperbolic Groups with Virtually Abelian Parabolic Subgroups
release_epo2n6yfs5bkzp675hbtrchnkm
by
Matthieu Dussaule,
Ilya Gekhtman,
Victor Gerasimov,
Leonid Potyagailo
Published
by Mathematisches Forschungsinstitut Oberwolfach.
2018 p03
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 ${\mathcal H}^n$, 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.
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