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

By Martin R. Bridson. 2009

68 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7098d24da069f0fb851e1719d7a972574e0a1ea8

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Citations in the corpus

Growth tight actions (2014, background): “When the genus of S is at least 3 then there does not exist a cocompact, CAT(0) Mod(S)-space [Bridson 2010].”
Constructing group actions on quasi-trees and applications to mapping class groups (2010, methodology): “Theorem F provides a sharp contrast to a result of Bridson [Bri10], who showed that in semi-simple actions of mapping class groups (of genus > 2) on complete CAT (0) spaces Dehn twists are always elliptic.”