Rigidity of mapping class group actions on S1

Mann, Kathryn; Wolff, Maxime

doi:10.2140/gt.2020.24.1211

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Rigidity of mapping class group actions on $S^1$

Kathryn Mann and Maxime Wolff

Geometry & Topology 24 (2020) 1211–1223

DOI: 10.2140/gt.2020.24.1211

Abstract

The mapping class group ${Mod}_{g, 1}$ of a surface with one marked point can be identified with an index two subgroup of $Aut (π_{1} Σ_{g})$ . For a surface of genus $g \geq 2$ , we show that any action of ${Mod}_{g, 1}$ on the circle is either semiconjugate to its natural faithful action on the Gromov boundary of $π_{1} Σ_{g}$ , or factors through a finite cyclic group. For $g \geq 3$ , all finite actions are trivial. This answers a question of Farb.

Keywords

mapping class group, Gromov boundary, surface group, rigidity, homeomorphisms of the circle, Euler class

Mathematical Subject Classification 2010

Primary: 57M60

Secondary: 20F34, 57M07

References

Publication

Received: 15 September 2018

Revised: 28 May 2019

Accepted: 11 October 2019

Published: 30 September 2020

Proposed: Jean-Pierre Otal

Seconded: John Lott, Ulrike Tillmann

Authors

Kathryn Mann
Department of Mathematics Cornell University Ithaca, NY United States

Maxime Wolff
Sorbonne Universités UPMC Univ. Paris 06 Institut de Mathématiques de Jussieu-Paris Rive Gauche UMR 7586 CNRS Univ. Paris Diderot Paris France

Volume 24, issue 3 (2020)