Martin David Palmer -- Preprint -- Polynomiality of surface braid and mapping class group representations
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Polynomiality of surface braid and mapping class group representations
with Arthur Soulié
(v1: 2023)

arXiv: abspdf
v1

Abstract

A wide family of homological representations of surface braid groups and mapping class groups of surfaces was developed in arXiv:1910.13423. These representations are naturally defined as functors on a category whose automorphism groups are the family of groups under consideration, and whose richer structure may be used to prove twisted homological stability results — subject to the condition that the functor is polynomial. We prove that many of these homological representation functors are polynomial, including those extending the Lawrence-Bigelow representations of the classical braid groups. In particular, we carry out general computations of the homological representation modules by using Borel-Moore homology and qualitative properties of the group actions. These polynomiality results also have applications for representation theoretic questions.

The annex file for this paper, containing additional details and calculations, is here.