Mapping class group representations from Drinfeld doubles of finite groups

Abstract

We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group [Formula: see text], 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 [Formula: see text] their restriction to the Torelli group is non-trivial if and only if [Formula: see text] is non-abelian.