Mapping class group representations from Drinfeld doubles of finite groups

TL;DR

This paper explores how mapping class group representations derived from the Drinfeld double of finite groups can be explicitly described and analyzed, revealing their finite images and non-triviality conditions related to the group's abelian property.

Contribution

It provides concrete descriptions of these representations in terms of finite group data and establishes their properties, including finiteness of images and Torelli group restrictions.

Findings

Representations have finite images.

Non-trivial Torelli group restriction iff G is non-abelian.

Concrete descriptions in terms of finite group data.

Abstract

We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group G, focusing on surfaces without marked points or with one marked point. We obtain concrete descriptions of such representations in terms of finite group data. This allows us to establish various properties of these representations. In particular we show that they have finite images, and that for surfaces of genus at least 3 their restriction to the Torelli group is non-trivial iff G is non-abelian.