#### 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
##### Abstract

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.

##### Keywords
random walk, Ancona inequalities, hierarchically hyperbolic group, relatively hyperbolic group
##### Mathematical Subject Classification 2010
Primary: 20P05, 20F65, 60J50
Secondary: 20F67, 31C35