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

#### Martin R. Bridson

#### Final edits 8 April 2009; to appear in CUP
volume in honour of Bill Harvey's 65th birthday:

"The Geometry of Riemann Surfaces", LMS Lecture Notes 368.

- This is a .pdf file
(13 pages, no figures)
Let $\S$
be an orientable surface of finite type
and let ${\rm{Mod}}(\Sigma)$ be its mapping class group. We consider actions of ${\rm{Mod}}(\Sigma)$ by semisimple
isometries on complete {\rm{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 ${\rm{Mod}}(\Sigma)$ 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 {\rm{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 {\rm{CAT}}$(0)$ space of dimension $18$.