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Property $R_{\infty}$ for groups with infinitely many ends

Francesco Fournier-Facio, Harry Iveson, Armando Martino, Wagner Sgobbi, Peter Wong

Abstract

We show that an accessible group with infinitely many ends has property $R_{\infty}$. That is, it has infinitely many twisted conjugacy classes for any twisting automorphism. We deduce that having property $R_{\infty}$ is undecidable amongst finitely presented groups. We also show that the same is true for a wide class of relatively hyperbolic groups, filling in some of the gaps in the literature. Specifically, we show that a non-elementary, finitely presented relatively hyperbolic group with finitely generated peripheral subgroups which are not themselves relatively hyperbolic, has property $R_{\infty}$.

Property $R_{\infty}$ for groups with infinitely many ends

Abstract

We show that an accessible group with infinitely many ends has property . That is, it has infinitely many twisted conjugacy classes for any twisting automorphism. We deduce that having property is undecidable amongst finitely presented groups. We also show that the same is true for a wide class of relatively hyperbolic groups, filling in some of the gaps in the literature. Specifically, we show that a non-elementary, finitely presented relatively hyperbolic group with finitely generated peripheral subgroups which are not themselves relatively hyperbolic, has property .

Paper Structure

This paper contains 20 sections, 50 theorems, 17 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Twisted Conjugacy and the Property
  3. Twisted Conjugacy
  4. Mapping Torus
  5. Trees and Group Actions
  6. Simplicial trees and
  7. Group actions on -trees
  8. Deformation spaces and reduced trees
  9. Irreducible actions on trees
  10. Actions and automorphisms
  11. Train Track Maps and free products
  12. Free factor systems
  13. for Groups with Infinitely Many Ends
  14. Accessible groups
  15. Relatively hyperbolic groups
  16. ...and 5 more sections

Key Result

Theorem 1.0.1

Any accessible group $G$ with infinitely many ends has property $R_{\infty}$. In particular, any finitely presented group with infinitely many ends has property $R_{\infty}$.

Figures (3)

  • Figure : On the axis: $\circ - \bullet - \circ - \bullet$
  • Figure : On the axis: $\circ - \bullet - \circ - \bullet$
  • Figure : Not on the axis: $\circ - \bullet - \bullet - \circ$

Theorems & Definitions (140)

  • Theorem 1.0.1
  • Corollary 1.0.0
  • Theorem 1.0.1
  • Remark
  • Corollary 1.0.0
  • Definition 2.1.1
  • Remark
  • Remark
  • Lemma 2.1.2
  • Definition 2.1.3: $R_{\infty}$
  • ...and 130 more