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Groups of order 64 and non-homeomorphic double Kodaira fibrations with the same biregular invariants
Francesco Polizzi, Pietro Sabatino
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Let Σ_b be a closed Riemann surface of genus b. We investigate finite quotients G of the pure braid group on two strands 𝖯₂(Σ_b) which do not factor through π₁(Σ_b ×Σ_b). Building on our previous work on some special systems of generators on finite groups that we called \emph{diagonal double Kodaira structures}, we prove that, if G has not order $32$, then |G| ≥64, and we completely classify the cases where equality holds. In the last section, as a geometric application of our algebraic results, we construct two $3$-dimensional families of double Kodaira fibrations having the same biregular invariants and the same Betti numbers but different fundamental group.
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