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Martin boundary covers Floyd boundary

MPS-Authors
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Gekhtman,  Michael
Max Planck Institute for Mathematics, Max Planck Society;

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Gerasimov,  Victor
Max Planck Institute for Mathematics, Max Planck Society;

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Potyagailo,  Leonid
Max Planck Institute for Mathematics, Max Planck Society;

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1708.02133.pdf
(Preprint), 647KB

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Citation

Gekhtman, M., Gerasimov, V., Potyagailo, L., & Yang, W. (2021). Martin boundary covers Floyd boundary. Inventiones mathematicae, 223(2), 759-809. doi:10.1007/s00222-020-01015-z.


Cite as: http://hdl.handle.net/21.11116/0000-0008-15AF-C
Abstract
For finitely supported random walks on finitely generated groups $G$ we prove that the identity map on $G$ extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. This yields new results for relatively hyperbolic groups. Our key estimate relates the Green and Floyd metrics, generalizing results of Ancona for random walks on hyperbolic groups and of Karlsson for quasigeodesics. We then apply these techniques to obtain some results concerning the harmonic measure on the limit sets of geometrically finite isometry groups of Gromov hyperbolic spaces. .