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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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