Any global fixed point in the low-dimensional deformation space of a surface group under pure mapping class group action is the trivial representation for genus at least three.
McCarthy,On the first cohomology group of cofinite subgroups in surface mapping class groups, Topology40(2001), no
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Global fixed point in low-dimensional surface group deformation space
Any global fixed point in the low-dimensional deformation space of a surface group under pure mapping class group action is the trivial representation for genus at least three.