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

Abstract

Let ${\mathsf{B}}_2({\Sigma }_g)$ 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 $\phi :{\mathsf{B}}_2({\Sigma }_g)\to G$, under the assumption that the quotient map $\phi$ does not factor through ${\pi }_1(\mathrm{Sy}{\mathrm{m}}^2{\Sigma }_g)$. We then apply our algebraic results to the geometric problem of constructing smooth surfaces of general type as Galois covers of $\mathrm{Sy}{\mathrm{m}}^2({\Sigma }_g)$ 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.

26 pages, 2 figures