On the dimension of CAT(0) spaces where mapping class groups act

TL;DR

This paper proves that the mapping class group of a closed orientable surface of genus g must fix a point when acting by semisimple isometries on any complete CAT(0) space of dimension less than g.

Contribution

It establishes a new fixed point property for mapping class groups acting on low-dimensional CAT(0) spaces, linking group actions to geometric dimension constraints.

Findings

Mapping class groups fix points in low-dimensional CAT(0) spaces

Fixed point property holds for spaces of dimension less than genus g

Provides new insights into the geometric actions of surface groups

Abstract

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 fixes a point.