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Unicorn paths and hyperfiniteness for the mapping class group

Part of:Ergodic theorySet theory

Published online by Cambridge University Press:  29 April 2021

Piotr Przytycki and
Marcin Sabok
Piotr Przytycki
Affiliation:
Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke St. W, Montreal, Quebec, Canada H3A 0B9; E-mail: piotr.przytycki@mcgill.ca; marcin.sabok@mcgill.ca.
Marcin Sabok
Affiliation:
Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke St. W, Montreal, Quebec, Canada H3A 0B9; E-mail: piotr.przytycki@mcgill.ca; marcin.sabok@mcgill.ca.

Abstract

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Let S be an orientable surface of finite type. Using Pho-on’s infinite unicorn paths, we prove the hyperfiniteness of orbit equivalence relations induced by the actions of the mapping class group of S on the Gromov boundaries of the arc graph and the curve graph of S. In the curve graph case, this strengthens the results of Hamenstädt and Kida that this action is universally amenable and that the mapping class group of S is exact.

MSC classification

Secondary:37A20: Orbit equivalence, cocycles, ergodic equivalence relations03E15: Descriptive set theory

Information

Type
Topology
Information
Forum of Mathematics, Sigma , Volume 9 , 2021, e36
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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