Global fixed point in low-dimensional surface group deformation space

Abstract: Under the natural action of the pure mapping class group of a surface, we show that any global fixed point in the low-dimensional deformation space of the fundamental group of the surface corresponds to the trivial representation, assuming the surface has genus greater than two. This result provides an alternative proof for a special case of a theorem by Landesman and Litt with a slight refinement. We also discuss a similar approach that may potentially lead to an alternative proof of the entirety of the theorem.