An embedding of the Morse boundary in the Martin boundary

Cordes, Matthew; Dussaule, Matthieu; Gekhtman, Ilya

doi:10.2140/agt.2022.22.1217

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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

DOI: 10.2140/agt.2022.22.1217

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

References

Publication

Received: 21 May 2020

Revised: 12 May 2021

Accepted: 1 July 2021

Published: 25 August 2022

Authors

Matthew Cordes
D-MATH ETH Zürich Zürich Switzerland

Matthieu Dussaule
Institut Denis Poisson Université de Tours Tours France

Ilya Gekhtman
Department of Mathematics Technion-Israel Institute of Technology Haifa Israel

Volume 22, issue 3 (2022)