Homological representations of low genus mapping class groups

2023/03/06 by Trent Lucas, Lucas, Trent
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2303.03552

openalex publication_date 2023/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a finite group G acting on a surface S, the centralizer of G in the mapping class group \textrmMod(S) has a natural representation given by its action on the homology H₁(S; ℚ). We consider the question of whether this representation has arithmetic image. Several authors have given positive and negative answers to this question. We give a complete answer when S has genus at most 3.

Related

• Arithmetic representations of mapping class groups
• Linear representations of the mapping class group of dimension at most 3g-3
• Representations of surface groups with finite mapping class group orbits
• Mapping class groups are not linear in positive characteristic
• Low-dimensional linear representations of mapping class groups