Volume 22, issue 3 (2022)

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An embedding of the Morse boundary in the Martin boundary

Matthew Cordes, Matthieu Dussaule and Ilya Gekhtman

Algebraic & Geometric Topology 22 (2022) 1217–1253

We construct a one-to-one continuous map from the Morse boundary of a hierarchically hyperbolic group to its Martin boundary. This construction is based on deviation inequalities generalizing Ancona’s work on hyperbolic groups. This provides a possibly new metrizable topology on the Morse boundary of such groups. We also prove that the Morse boundary has measure 0 with respect to the harmonic measure unless the group is hyperbolic.

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random walk, Ancona inequalities, hierarchically hyperbolic group, relatively hyperbolic group
Mathematical Subject Classification 2010
Primary: 20P05, 20F65, 60J50
Secondary: 20F67, 31C35
Received: 21 May 2020
Revised: 12 May 2021
Accepted: 1 July 2021
Published: 25 August 2022
Matthew Cordes
ETH Zürich
Matthieu Dussaule
Institut Denis Poisson
Université de Tours
Ilya Gekhtman
Department of Mathematics
Technion-Israel Institute of Technology