Gassner and Burau representations over ℤ_p-modules

2022/08/25 by Vasudha Bharathram, Bharathram, Vasudha
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2208.12378

Abstract

We study two classical representations of Artin's braid group and their modulo p reductions. We use topological methods to show that the Gassner representation τ_n: B_n\toGL_n(ℤ[t₁^{± 1}, …, t_n^{± 1}]) is faithful for all n, and furthermore that it is faithful modulo p for all integers p>1. We then give a novel proof that the Burau representation of B₃ is faithful modulo p for all p>1, and suggest applications to the modulo p Burau representation for higher braid groups.

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