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Groups of order 64 and non-homeomorphic double Kodaira fibrations with the same biregular invariants

Francesco Polizzi, Pietro Sabatino

Abstract

Let $Σ_b$ be a closed Riemann surface of genus $b$. We investigate finite quotients $G$ of the pure braid group on two strands $\mathsf{P}_2(Σ_b)$ which do not factor through $π_1(Σ_b \times Σ_b)$. 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 $|G| \geq 64$, 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.

Groups of order 64 and non-homeomorphic double Kodaira fibrations with the same biregular invariants

Abstract

Let be a closed Riemann surface of genus . We investigate finite quotients of the pure braid group on two strands which do not factor through . 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 has not order , then , and we completely classify the cases where equality holds. In the last section, as a geometric application of our algebraic results, we construct two -dimensional families of double Kodaira fibrations having the same biregular invariants and the same Betti numbers but different fundamental group.

Paper Structure

This paper contains 20 sections, 24 theorems, 54 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Group-theoretical preliminaries I: CCT groups
  3. CCT-groups
  4. The case |G|=36
  5. The case |G|=40
  6. The case |G|=48
  7. The case |G|=54
  8. The case |G|=56|
  9. The case |G|=60
  10. Group-theoretical preliminaries II: monolithic groups
  11. Diagonal double Kodaira structures and diagonal double Kodaira fibrations
  12. Diagonal double Kodaira structures on groups of order 64
  13. The case $G=(64, 199)$
  14. The case $G=(64, 200)$
  15. The case $G=(64, 201)$
  16. ...and 5 more sections

Key Result

Proposition 1.2

For a finite group $G$, the following properties are equivalent.

Theorems & Definitions (43)

  • Definition 1
  • Definition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 1.5
  • proof
  • Proposition 1.6
  • proof
  • Proposition 1.7
  • ...and 33 more