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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2310.10401 |
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| _version_ | 1866910701322240000 |
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| author | Menet, Gabrielle Nguyen, Duc-Manh |
| author_facet | Menet, Gabrielle Nguyen, Duc-Manh |
| contents | Let $d \geq 2$ and $n\geq 3$ be two natural numbers. Given any sequence $κ=(k_1,\dots,k_n) \in \mathbb{Z}^n$ 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 $ρ_d$ of $\mathrm{PB}_n$ into a linear algebraic group defined over $\mathbb{Q}$. In a sense, $ρ_d$ is primitive with respect to the parameters $d$ and $κ$. The first main result of this paper is a criterion for the Zariski closure of the image of $ρ_d$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_10401 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Representations of braid groups via cyclic covers of the sphere: Zariski closure and arithmeticity Menet, Gabrielle Nguyen, Duc-Manh Geometric Topology Complex Variables Group Theory 57K20 (Primary) 20C99 (Secondary) Let $d \geq 2$ and $n\geq 3$ be two natural numbers. Given any sequence $κ=(k_1,\dots,k_n) \in \mathbb{Z}^n$ 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 $ρ_d$ of $\mathrm{PB}_n$ into a linear algebraic group defined over $\mathbb{Q}$. In a sense, $ρ_d$ is primitive with respect to the parameters $d$ and $κ$. The first main result of this paper is a criterion for the Zariski closure of the image of $ρ_d$ 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. |
| title | Representations of braid groups via cyclic covers of the sphere: Zariski closure and arithmeticity |
| topic | Geometric Topology Complex Variables Group Theory 57K20 (Primary) 20C99 (Secondary) |
| url | https://arxiv.org/abs/2310.10401 |