Extra-special quotients of surface braid groups and double Kodaira fibrations with small signature

DOI10.1007/S10711-022-00720-8arXiv2102.04963MaRDI QIDQ6360229

Pietro Sabatino, Francesco Polizzi

Publication date: 9 February 2021

Abstract: We study some special systems of generators on finite groups, introduced in previous work by the first author and called "diagonal double Kodaira structures", in order to investigate non-abelian, finite quotients of the pure braid group on two strands $mathsf{P}_2(Sigm{a}_b)$, where $Sigm{a}_b$ is a closed Riemann surface of genus $b$. In particular, we prove that, if a finite group $G$ admits a diagonal double Kodaira structure, then $|G|geq32$, and equality holds if and only if $G$ is extra-special. In the last section, as a geometrical application of our algebraic results, we construct two $3$-dimensional families of double Kodaira fibrations having signature $16$. Such surfaces are different from the ones recently constructed by Lee, L"onne and Rollenske and, as far as we know, they provide the first examples of positive-dimensional families of double Kodaira fibrations with small signature.

Mathematics Subject Classification ID

Surfaces of general type (14J29) Finite nilpotent groups, (p)-groups (20D15) Special surfaces (14J25)

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