Global fixed point in low-dimensional surface group deformation space

TL;DR

The paper studies global fixed points of the pure mapping class group action on low-dimensional deformation spaces $X_r$ of surface groups for genus $g \ge 3$. It constructs a linear representation $\rho_\phi$ from a fixed point $\phi$ and uses low-dimensional classification results for $\mathcal{P}\mathcal{M}_g^n$ representations, together with finite-orbit analyses, to show that such fixed points must be trivial when $r \le \sqrt{2g}$; a genus-2 exception with $r=2$ yields finite image. This provides a shorter alternative proof of a special case of Landesman–Litt and suggests a pathway toward proving the full theorem via extensions to finite-index subgroups and deeper representation-theoretic constraints. The work connects to broader conjectures (Ivanov, Putman–Wieland) and highlights open problems in the low-dimensional representation theory of finite-index subgroups of mapping class groups. Overall, it offers a method to translate fixed-point questions into finite-dimensional linear-algebra constraints, with potential implications for the structure of deformation spaces and monodromy representations.

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.

Theorems & Definitions (12)

• Theorem A: Landesman--Litt
• Theorem B
• Theorem C
• Lemma 1.1
• proof : Proof of Lemma \ref{['lem']}
• Remark 1.2
• Theorem 2.1
• Corollary 2.2
• Remark 2.3
• Theorem 2.4: BKMS