Homological representations of families of groups
Arthur Soulié (University of Glasgow)
Wednesday 9th December, 2020 16:00-17:00 Maths 311B
Many families of groups, such as braid groups, have a representation theory of wild type, in the sense that there is no known classification schema to organize the representations. However, using actions on the homology groups of the coverings of some associated spaces, there are systematic procedures to construct linear representations for such families of groups, which help to understand their representation theory.
I will present a unified functorial construction of homological representations for these families of groups. This general method is particularly suitable to generate new families of representations of motion groups such as braid groups on surfaces or loop braid groups. For instance, this construction provides the family of Lawrence-Bigelow representations for braid groups. We will also discuss irreducibility results for the obtained representations. Finally, general notions of polynomiality on functors are a useful tool to classify these representations and allow to prove some twisted homological stability results: polynomiality results can be proved for some of the homological representations, in particular the Lawrence-Bigelow representations.