Images of quantum representations of mapping class groups and Dupont-Guichardet-Wigner quasi-homomorphisms

TL;DR

This paper investigates the properties of quantum representations of mapping class groups, revealing their cohomological features and conditions under which their images resemble higher rank lattices, using quasi-homomorphisms derived from pseudo-unitary groups.

Contribution

It establishes new results on the structure and cohomology of quantum representations of mapping class groups, introducing quasi-homomorphisms extending previous work.

Findings

Images are not isomorphic to higher rank lattices or kernels have many normal generators.

Images have nontrivial 2-cohomology at small levels.

Constructs quasi-homomorphisms from pseudo-unitary groups to study these properties.

Abstract

We prove that either the images of the mapping class groups by quantum representations are not isomorphic to higher rank lattices or else the kernels have a large number of normal generators. Further we show that the images of the mapping class groups have nontrivial 2-cohomology, at least for small levels. For this purpose we considered a series of quasi-homomorphisms on mapping class groups extending previous work of Barge and Ghys and of Gambaudo and Ghys. These quasi-homomorphisms are pull-backs of the Dupont-Guichardet-Wigner quasi-homomorphisms on pseudo-unitary groups along quantum representations.