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

Kathryn Mann and Maxime Wolff

Geometry & Topology 24 (2020) 1211–1223

The mapping class group Modg,1 of a surface with one marked point can be identified with an index two subgroup of Aut(π1Σg). For a surface of genus g 2, we show that any action of Modg,1 on the circle is either semiconjugate to its natural faithful action on the Gromov boundary of π1Σg, or factors through a finite cyclic group. For g 3, all finite actions are trivial. This answers a question of Farb.

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mapping class group, Gromov boundary, surface group, rigidity, homeomorphisms of the circle, Euler class
Mathematical Subject Classification 2010
Primary: 57M60
Secondary: 20F34, 57M07
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
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
Univ. Paris Diderot