Image of the Burau representation at $d$-th roots of unity


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 $n\geq 2d$.


T. N. Venkataramana

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai, 40005, India