Semisimple actions of mapping class groups on CAT(0) spaces

TL;DR

This paper investigates semisimple isometric actions of mapping class groups on CAT(0) spaces, revealing fixed point properties for high-genus surfaces and providing examples of proper actions in genus 2.

Contribution

It establishes fixed point results for high-genus mapping class groups and constructs explicit proper actions for genus 2 groups on low-dimensional CAT(0) spaces.

Findings

Dehn twists act elliptically in high-genus actions

High-genus mapping class groups fix points in low-dimensional CAT(0) spaces

Genus 2 mapping class group acts properly on an 18-dimensional CAT(0) space

Abstract

Let S be an orientable surface of finite type and let Mod(S) be its mapping class group. We consider actions of Mod(S) by semisimple isometries on complete CAT(0) spaces. If the genus of S is at least 3, then in any such action all Dehn twists act as elliptic isometries. The action of Mod(S) on the completion of Teichm\"uller space with the Weil-Petersson metric shows that there are interesting actions of this type. Whenever the mapping class group of a closed orientable surface of genus g acts by semisimple isometries on a complete CAT(0) space of dimension less than g it must fix a point. The mapping class group of a closed surface of genus 2 acts properly by semisimple isometries on a complete CAT(0) space of dimension 18.