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

2026/07/20 by Massimiliano Alessandro, Michelangelo Migliano, Francesco Polizzi
#math.GR #math.AG

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Abstract

Let B₂(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 B₂(Σ_g) → G, under the assumption that the quotient map φ does not factor through π₁(Sym²Σ_g). We then apply our algebraic results to the geometric problem of constructing smooth surfaces of general type as Galois covers of Sym²(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 K²=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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