Speaker: Arthur Soulié
Title: On homological representations of families of motion groups and mapping class groups
Abstract: I will describe a general construction of homological representations for families of motion groups or mapping class groups, including the families of braid groups, surface braid groups and loop braid groups. This recovers the well-known constructions of Lawrence-Bigelow, and in this sense it unifies these constructions. I will also discuss indecomposability and irreducibility of these representations. In particular, any Lawrence-Bigelow representation is indecomposable.
The construction is moreover “global” in the sense that, for each dimension d, it is a functor on a category whose automorphism groups are all d-dimensional motion groups and mapping class groups, and which also carries a richer structure. Using this richer structure, I will discuss polynomiality of these families of representations, and use this to prove twisted homological stability for the braid groups with coefficients in any one of the Lawrence-Bigelow representations.
All this represents a joint work with Martin Palmer.
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