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Extending finite free actions of surfaces

Rubén A. Hidalgo

Abstract

We prove the existence of finite groups of orientation-preserving homeomorphisms of some closed orientable surface $S$ that act freely and which extends as a group of homeomorphisms of some compact orientable $3$-manifold with boundary $S$, but which cannot extend to a handlebody.

Extending finite free actions of surfaces

Abstract

We prove the existence of finite groups of orientation-preserving homeomorphisms of some closed orientable surface that act freely and which extends as a group of homeomorphisms of some compact orientable -manifold with boundary , but which cannot extend to a handlebody.
Paper Structure (22 sections, 14 theorems, 22 equations, 2 figures)

This paper contains 22 sections, 14 theorems, 22 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Actions on handlebodies and Schottky groups
  4. The Bogomolov multiplier
  5. Samperton's theorem
  6. Examples of groups which extends to handlebodies
  7. A special set of loops
  8. Automorphism group of the genus two fundamental group
  9. The collection ${\mathfrak C}$
  10. A simple condition for extendability of free actions to handlebodies
  11. Necessary and sufficient conditions: The case $\gamma=2$
  12. Necessary and sufficient conditions: The case $\gamma \geq 3$
  13. Some extra remarks
  14. Finite groups acting freely which extend but not to handlebodies
  15. An algorithm to construct free actions that extend but not to handlebodies
  16. ...and 7 more sections

Key Result

Theorem 1

Let $K$ be a virtual (extended) Schottky group containing as a finite index normal subgroup a Schottky group $\Gamma$ of rank $g \geq 2$. Assume that the group $K/\Gamma$ does not contain dihedral subgroups and that it acts freely on $\Omega/\Gamma$, where $\Omega$ is the region of discontinuity. Th

Figures (2)

  • Figure 1: The group $F$ seen as $\pi_{1}(R)$
  • Figure 2: Tuples $[p,q,r]$, where $5 \leq q <p \leq 23$, in Example \ref{['Sec:Ejemplo1']} with a corresponding $w \in {\mathfrak C}_{0} \cap \ker(\theta)$

Theorems & Definitions (21)

  • Theorem 1: H3
  • Lemma 1
  • Theorem 2: Samperton Samperton
  • Corollary 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 2
  • Theorem 3
  • ...and 11 more