On finite quotients of surface braid groups having order at most $127$

TL;DR

This work classifies admissible finite quotients of the pure surface braid group $\mathsf{P}_2(\Sigma_b)$ of size at most $127$, revealing that only the genus-2, double-cover type $(b,n)=(2,2)$ occur in this range. By translating topological data into Diophantine-like combinatorial structures (diagonal double Kodaira structures) on finite groups and employing GAP4 computations, the authors isolate exactly 11 admissible quotients among 1036 non-abelian groups of small order, with two new order-96 cases and detailed lifting from smaller extraspecial quotients. The study develops a robust framework of prestructures and monolithic/CCT-group criteria to rule out many candidates, and it establishes a precise list of groups that realize admissible quotients or prestructures under 127 elements. The results illuminate the scarcity of finite quotients realizing double Kodaira surface data and raise directions for higher-genus generalizations and a fuller classification problem for admissible braid quotients.

Abstract

Let $Σ_b$ be a compact Riemann surface of genus $b \geq 2$ and let $\mathsf{P}_2(Σ_b)=π_1(Σ_b \times Σ_b - Δ)$ be the corresponding pure braid group on two strands. A finite quotient $\varphi \colon \mathsf{P}_2(Σ_b) \to G$ is called "admissible" if $\varphi$ does not factor through $π_1(Σ_b \times Σ_b)$. In this work we classify all admissible quotients of $\mathsf{P}_2(Σ_b)$ such that $|G| \leq 127$.

Figures (2)

• Figure 1: The pure braids $\rho_{1j}, \, \tau_{1j}$, $\rho_{2j}, \, \tau_{2j}$ on $\Sigma_b$
• Figure 2: The pure braid $A_{12}$ on $\Sigma_b$

Theorems & Definitions (45)

• Definition 2.1
• Proposition 2.2
• Remark 2.3
• Proposition 2.4
• Proposition 2.5
• proof
• Example 2.6
• Example 2.7
• Example 2.8
• Proposition 2.9