Global fixed point in low-dimensional surface group deformation space

2025/10/07 by Kasahara, Yasushi
#57K20 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2510.05638

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.

Citations

• On the uniqueness of the Prym map
• Canonical representations of surface groups
• Crossed homomorphisms and low dimensional representations of mapping class groups of surfaces
• Representations of surface groups with finite mapping class group orbits
• On visualization of the linearity problem for mapping class groups of surfaces
• Low-dimensional linear representations of mapping class groups
• Triviality of some representations of MCG(S_g) in GL(n,C), Diff(S²)\n and Homeo(T²)
• Low-dimensional homology groups of mapping class groups: a survey

Related

• A Superficial Working Guide to Deformations and Moduli
• Surface classification and local and global fundamental groups, I
• Low-dimensional linear representations of mapping class groups
• Finite simple groups acting with fixity 3 and their occurrence as groups of automorphisms of Riemann surfaces (extended version)
• A Fixed Point Theorem for Deformation Spaces of G-trees