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The 2-rank of finite groups acting on hyperelliptic 3-manifolds

Max Leopold Frisch Sbarra, Mattia Mecchia

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 \textit{hyperelliptic} as the manifolds admitting such an action. We prove that the sectional 2-rank of a finite group acting on a 3-manifold and containing a hyperelliptic involution with fixed-point set with two components has sectional 2-rank at most four; this upper bound is sharp. The cases where the hyperelliptic involution has a fixed-point set with a number of components different from 2 have been already considered in literature. Our result completes the analysis and we obtain some general results where the number of the components of the fixed-point set is not fixed. In particular, we obtain that a finite group acting on a 3-manifold and containing a hyperelliptic involution has 2-rank at most four, and four is the best possible upper bound. Finally, we restrict to the basic case of simple groups acting on hyperelliptic 3-manifolds: we use our result about the sectional 2-rank to prove that a simple group containing a hyperelliptic involution is isomorphic to $PSL(2,q)$ for some odd prime power $q$, or to one of four other small simple groups.

The 2-rank of finite groups acting on hyperelliptic 3-manifolds

Abstract

We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to Such involutions are called \textit{hyperelliptic} as the manifolds admitting such an action. We prove that the sectional 2-rank of a finite group acting on a 3-manifold and containing a hyperelliptic involution with fixed-point set with two components has sectional 2-rank at most four; this upper bound is sharp. The cases where the hyperelliptic involution has a fixed-point set with a number of components different from 2 have been already considered in literature. Our result completes the analysis and we obtain some general results where the number of the components of the fixed-point set is not fixed. In particular, we obtain that a finite group acting on a 3-manifold and containing a hyperelliptic involution has 2-rank at most four, and four is the best possible upper bound. Finally, we restrict to the basic case of simple groups acting on hyperelliptic 3-manifolds: we use our result about the sectional 2-rank to prove that a simple group containing a hyperelliptic involution is isomorphic to for some odd prime power , or to one of four other small simple groups.
Paper Structure (4 sections, 13 theorems, 20 equations, 3 figures)

This paper contains 4 sections, 13 theorems, 20 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries results
  3. Proof of Theorem
  4. Simple groups

Key Result

Theorem 1.1

Let $M$ be a smooth, closed and orientable 3-manifold, and let $G$ be a finite group which acts on $M$ smoothly and orientation-preservingly. If $G$ contains a hyperelliptic involution whose fixed-point set has two components, then the sectional 2-rank of $G$ is at most 4.

Figures (3)

  • Figure 1: First possibility. The fixed-point set of $h$ (red), $f$ (blue) and $fh'$ (black).
  • Figure 2: Second possibility. The fixed-point set of $h$ (red), $f$ (blue) and $fh$ (black).
  • Figure 3: The three possible combinatorial structures of $\Gamma$; the projection of the fixed-point sets of $t,$$f^2,$ and $tf^2$ are respectively black, red and blue

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Remark 1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 12 more