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

Summary

Abstract

Let Σ be an orientable surface of finite type and let Mod(Σ) be its mapping class group. We consider actions of Mod(Σ) by semisimple isometries on complete CAT(0) spaces. If the genus of Σ is at least 3, then in any such action all Dehn twists act as elliptic isometries. The action of Mod(Σ) on the completion of Teichmüller 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.

Introduction

This article concerns actions of mapping class groups by isometries on complete CAT(0) spaces. It records the contents of the lecture that I gave at Bill Harvey's 65th birthday conference at Anogia, Crete in July 2007.

A CAT(0) space is a geodesic metric space in which each geodesic triangle is no fatter than a triangle in the Euclidean plane that has the same edge lengths (see Definition 2.1). Classical examples include complete 1-connected Riemannian manifolds with non-positive sectional curvature and metric trees. The isometries of a CAT(0) space X divide naturally into two classes: the semisimple isometries are those for which there exists x₀ ∈ X such that d(γ·x₀, x₀)= |γ| where |γ| := inf{d(γ·y, y) | y ∈ X}; the remaining isometries are said to be parabolic. Semisimple isometries are further divided into hyperbolics, for which |γ| > 0, and elliptics, which have fixed points.