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RESEARCH PRODUCT

MR 3219513 Reviewed Venkataramana T. N. Monodromy of cyclic coverings of the projective line. Invent. Math., 197 (2014), 1–-45. (Reviewer Francesca Vetro) 14H30 (14D05 22E40 20F36)

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Let $d \geq 2$ and $n \geq 1$ be integers and $P_{n+1}$ be the pure braid group on $n + 1$ strands. In this paper, the author studies the image of $P_{n+1}$ under the monodromy action on the homology of a cyclic covering of degree $d$ of the projective line. More precisely, let $k_{1}, \ldots, k_{n + 1}$ be integers such that $1 \leq k_{i} \leq d - 1$ and gcd$(k_{i}, d) = 1$ for each $i$. Moreover, let $a_{1}, \ldots, a_{n + 1}$ be distinct points of the complex plane and $C$ be the space of points in $\mathbb{C}^{n + 1}$ with all distinct coordinates. Let us denote by $X_{a, k}$ the affine curve defined by the equation $$ y^{d} = (x - a_{1})^{k_{1}} (x - a_{2})^{k_{2}} \cdots (x - a_{n +1})^{k_{n + 1}}$$ with $y \neq 0$ and $x \neq a_{1}, \ldots, a_{n + 1}$. The author observes that the curve $X_{a, k}$ is a compact Riemann surface $X_{a, k}^{\star}$ minus a finite set of punctures and as the point $a = (a_{1}, \ldots, a_{n + 1}) \in C$ varies, one obtains a family $F^{\star}$ of compact Riemann surfaces $X_{a, k}^{\star}$ fibering over $C$. Since $\pi_{1} (C)$ is $P_{n + 1}$, the fibration $F^{\star} \rightarrow C$ yields a monodromy representation $$ \rho^{\star}_{M} (k, d): P_{n + 1} \rightarrow GL( H_{1}(X_{a, k}^{\star}, \mathbb{Z}))$$ of $P_{n+1}$ on the integral homology of the fiber $X_{a, k}^{\star}$. The main result of the paper is given by the following theorem. \vspace{1ex} Theorem 1: \textit{Suppose that $n \geq 2d$. Then the image $ \rho^{\star}_{M} (k, d) (P_{n + 1} )$ of the monodromy representation $\rho^{\star}_{M} (k, d)$ of $P_{n + 1} $ is an arithmetic group. Moreover, the monodromy group is (up to finite index) a product of irreducible lattices each of which has $\mathbb{Q}$-rank at least two.} \vspace{1ex} We note that the author get the previous result by proving the arithmeticity of the images of certain representations of the pure braid group $P_{n + 1} $.