Representations of braid groups via cyclic covers of the sphere: Zariski closure and arithmeticity

Abstract

Let $d\ge 2$ and $n\ge 3$ be two natural numbers. Given any sequence $\kappa =(k_1,\dots ,k_n)\in {\mathbb{Z}}^n$ such that $\gcd (k_1,\dots ,k_n,d)=1$, we consider the family of Riemann surfaces obtained from the plane curves defined by $y^d={\prod }_{i=1}^n(x-b_i{)}^{k_i}$, where $\{b_1,\dots ,b_n\}$ are $n$ distinct points in $\mathbb{C}$. The monodromy of the cohomology of the fibers of this family provides us with a representation of the pure braid group ${\mathrm{PB}}_n$ into some symplectic group. By restricting to a specific subspace in the cohomology of the fibers, we obtain a representation ${\rho }_d$ of ${\mathrm{PB}}_n$ into a linear algebraic group defined over $\mathbb{Q}$. In a sense, ${\rho }_d$ is primitive with respect to the parameters $d$ and $\kappa$. The first main result of this paper is a criterion for the Zariski closure of the image of ${\rho }_d$ to be maximal, and the second main result is a criterion for the image to be an arithmetic lattice in the target group. The latter generalizes previous results of Venkataramana and gives an answer to a question by McMullen.

Statement of Theorem C corrected, 50 pages