Linear representations of the mapping class group of dimension at most 3g-3

2025/07/15 by Kaufmann, Julian, Salter, Nick, Zhang, Zhong +1 · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2507.11365

Abstract

We classify representations of the mapping class group of a surface of genus g (with at most one puncture or boundary component) up to dimension 3g-3. Any such representation is the direct sum of a representation in dimension 2g or 2g+1 (given as the action on the (co)homology of the surface or its unit tangent bundle) with a trivial representation. As a corollary, any linear system on the moduli space of Riemann surfaces of genus g in this range is of algebro-geometric origin.

Citations

• Lower bounds for faithful linear representations of subgroups of the mapping class group
• Motives, mapping class groups, and monodromy
• On the uniqueness of the Prym map
• Totally symmetric sets in the general linear group
• Crossed homomorphisms and low dimensional representations of mapping class groups of surfaces
• Global rigidity of the period mapping
• Effective and Infinite-Rank Superrigidity in the Context of Representation Stability
• Aspects of non positive curvature for linear groups with no infinite order unipotents
• Low-dimensional linear representations of mapping class groups

Cited by

• Lower bounds for faithful linear representations of subgroups of the mapping class group

Related

• Low-dimensional linear representations of mapping class groups
• The symplectic representation of the mapping class group is unique
• Crossed homomorphisms and low dimensional representations of mapping class groups of surfaces
• The hyperelliptic mapping class group of a nonorientable surface of genus g≥ 4 has a faithful representation into GL(g²-1,ℝ)
• Mapping class groups are not linear in positive characteristic