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arXiv · 2607.18493

Finite quotients of full surface braid groups and complex surfaces of general type: cyclic, dihedral, and extra-special quotients

Massimiliano Alessandro·Michelangelo Migliano·Francesco Polizzi
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Abstract

Let $\mathsf{B}_2(Σ_g)$ be the full braid group on two strings on a compact Riemann surface of genus $g$. We compute the number of finite cyclic, dihedral and extra-special quotients $φ\colon \mathsf{B}_2(Σ_g) \to G$, under the assumption that the quotient map $φ$ does not factor through $π_1(\operatorname{Sym^2}Σ_g)$. We then apply our algebraic results to the geometric problem of constructing smooth surfaces of general type as Galois covers of $\operatorname{Sym^2}(Σ_g)$ branched on the diagonal. In particular, we construct two $3$-dimensional families of minimal surfaces of general type with $p_g=7$, $q=4$ and $K^2=32$ such that members of different families have the same biregular invariants and the same Betti numbers, but different torsion part for the first homology group.

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Massimiliano Alessandro, Michelangelo Migliano, Francesco Polizzi. 2026-07-20. Finite quotients of full surface braid groups and complex surfaces of general type: cyclic, dihedral, and extra-special quotients. https://arxiv.org/abs/2607.18493

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