Actions of mapping class groups

TL;DR

This paper explores the rigidity properties of mapping class group actions on various geometric and topological spaces, detailing key results and open problems in the field.

Contribution

It provides a comprehensive overview of four major rigidity results concerning actions on spaces of foliations and laminations, and discusses future research directions.

Findings

Rigidity results for actions on Thurston's sphere of projective foliations

Rigidity of actions on the space of unmeasured foliations

Results on the reduced Bers boundary and geodesic laminations

Abstract

This paper has three parts. The first part is a general introduction to rigidity and to rigid actions of mapping class group actions on various spaces. In the second part, we describe in detail four rigidity results that concern actions of mapping class groups on spaces of foliations and of laminations, namely, Thurston's sphere of projective foliations equipped with its projective piecewise-linear structure, the space of unmeasured foliations equipped with the quotient topology, the reduced Bers boundary, and the space of geodesic laminations equipped with the Thurston topology. In the third part, we present some perspectives and open problems on other actions of mapping class groups. The paper will appear in the Handbook of Group actions, vol. I (ed. L. Ji, A. Papadopoulos and S.-T. Yau), Higher Eucation Press and International Press.