The pro-nilpotent Lawrence-Krammer-Bigelow representation
Moduli and Friends seminar
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Thursday 28 July 2022
Sala 309-310 "Gheorghe Vranceanu"

Arthur Soulié (University of Glasgow)The pro-nilpotent Lawrence-Krammer-Bigelow representation

The Lawrence-Bigelow representations are an important family of representations of the braid groups. The first is the Burau representation and the second is the Lawrence-Krammer-Bigelow representation, which was used by Bigelow and Krammer to prove that the braid groups are linear.
In a joint work with Martin Palmer (arxiv:1910.13423), we introduced a general setting into which this construction naturally belongs, which produces a wide family of representations of motion groups, such as surface braid groups and loop braid groups). Describing this construction and how it recovers the Lawrence-Bigelow representations will be the starting point of my talk.
A key tool in this procedure is the use of the lower central series (LCS) of some associated "mixed" motion groups (e.g. the second term of the LCS of the partitioned braid groups on two blocks for the braid groups for the Lawrence-Bigelow representations). The length of these LCS's (and thus the relevance of considering all their terms) have been intensively studied in a joint work with Martin Palmer and Jacques Darné (arxiv:2201.03542), whose results will be briefly summed up.
As long as the lower central series of an associated "mixed" motion group has infinite length with some further conditions, the general construction naturally produces in some cases infinitely many new families of representations of motion groups, which assemble into "nilpotent towers". This represents a joint work in progress with Martin Palmer. In particular, we introduce a pro-nilpotent version of the Lawrence-Krammer-Bigelow representations (i.e. a nilpotent tower such that the family of the Lawrence-Krammer-Bigelow representations is the bottom layer of such a tower). I will discuss this case, its first properties and the analogous cases for surface braid groups and loop braid groups.

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