Crossed homomorphisms and low dimensional representations of mapping class groups of surfaces

2022/02/13 by Yasushi Kasahara, Kasahara, Yasushi · 4 citations
Mathematics · #Geometric and Algebraic Topology #Advanced Algebra and Geometry #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2202.06368

Abstract

We continue the study of low dimensional linear representations of mapping class groups of surfaces initiated by Franks--Handel and Korkmaz. We consider (2g+1)-dimensional complex linear representations of the pure mapping class groups of compact orientable surfaces of genus g. We give a complete classification of such representations for g ≥ 7 up to conjugation, in terms of certain twisted 1-cohomology groups of the mapping class groups. A new ingredient is to use the computation of a related twisted 1-cohomology group by Morita. The classification result implies in particular that there are no irreducible linear representations of dimension 2g+1 for g ≥ 7, which marks a contrast with the case g=2.

Cited by

• Deformations of crossed homomorphisms on Lie groups
• Lower bounds for faithful linear representations of subgroups of the mapping class group
• Global fixed point in low-dimensional surface group deformation space
• Linear representations of the mapping class group of dimension at most 3g-3

Related

• Low-dimensional linear representations of mapping class groups
• Linear representations of the mapping class group of dimension at most 3g-3
• Lower bounds for faithful linear representations of subgroups of the mapping class group
• The symplectic representation of the mapping class group is unique
• Mapping class groups are not linear in positive characteristic