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

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

2009-08-05

arxiv.org/abs/0908.0685v1

Mathematics

Geometric Topology

To appear in "The Geometry of Riemann Surfaces", LMS Lecture Notes 368. Dedicated to Bill Harvey on his 65th birthday. 12 page

Scientific paper

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.

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