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Hyperfiniteness of boundary actions of cubulated hyperbolic groups

Part of: Set theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  25 March 2019

JINGYIN HUANG ,
MARCIN SABOK  and
FORTE SHINKO
JINGYIN HUANG
Affiliation:
McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street W, Montreal, QC, H3A 0B9Canada email jingyin.huang@mail.mcgill.ca, marcin.sabok@mcgill.ca, forte.shinko@mail.mcgill.ca
MARCIN SABOK
Affiliation:
McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street W, Montreal, QC, H3A 0B9Canada email jingyin.huang@mail.mcgill.ca, marcin.sabok@mcgill.ca, forte.shinko@mail.mcgill.ca Instytut Matematyczny PAN, Śniadeckich 8, 00-656Warszawa, Poland
FORTE SHINKO
Affiliation:
McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street W, Montreal, QC, H3A 0B9Canada email jingyin.huang@mail.mcgill.ca, marcin.sabok@mcgill.ca, forte.shinko@mail.mcgill.ca
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Abstract

We show that if a hyperbolic group acts geometrically on a CAT(0) cube complex, then the induced boundary action is hyperfinite. This means that for a cubulated hyperbolic group, the natural action on its Gromov boundary is hyperfinite, which generalizes an old result of Dougherty, Jackson and Kechris for the free group case.

Keywords

group actionstopological dynamicshyperbolic groupshyperfinite equivalence relationsGromov boundary

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 20F65: Geometric group theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 9 , September 2020 , pp. 2453 - 2466
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Creative Commons
This is a work of the U.S. Government and is not subject to copyright protection in the United States.
Copyright
© Cambridge University Press, 2019

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