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Image of the Burau representation at \(d\)-th roots of unity. (English) Zbl 1345.20051
Summary: We show that the image of the braid group under the monodromy action on the homology of a cyclic covering of degree \(d\) of the projective line is an arithmetic group provided the number of branch points is sufficiently large compared to the degree \(d\). This is deduced by proving the arithmeticity of the image of the braid group on \(n+1\) letters under the Burau representation evaluated at \(d\)-th roots of unity when \(n\geq 2d\).

MSC:
20F36 Braid groups; Artin groups
20G30 Linear algebraic groups over global fields and their integers
20C15 Ordinary representations and characters
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