Finite groups acting on hyperelliptic 3-manifolds

TL;DR

This paper classifies finite groups acting on hyperelliptic 3-manifolds, especially those with involutions having multiple fixed points, revealing that such groups are mostly isomorphic to specific small simple groups or PSL(2,q).

Contribution

It characterizes the structure of finite groups with hyperelliptic involutions on 3-manifolds, identifying the possible simple groups involved.

Findings

Simple groups with such involutions are isomorphic to PSL(2,q) or one of four small simple groups.

Provides classification of groups acting on hyperelliptic 3-manifolds with fixed-point set having more than two components.

Establishes constraints on the algebraic structure of groups related to hyperelliptic involutions.

Abstract

We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to $S^3.$ Such involutions are called hyperelliptic as the manifolds admitting such an action. We consider finite groups acting on 3-manifolds and containing hyperelliptic involutions whose fixed-point set has $r>2$ components. In particular we prove that a simple group containing such an involution is isomorphic to $PSL(2,q)$ for some odd prime power $q$, or to one of four other small simple groups.