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Homeomorphism types of Floyd boundaries of infinite-ended groups

Subhajit Chakraborty, Ravi Tomar

Abstract

Suppose $G$ is a finitely generated infinite group, and $\mathcal G$ is a graph of groups decomposition of $G$ such that the edge groups are finite. This paper establishes that the topology of the Floyd boundary of $G$ is uniquely determined by the topology of the Floyd boundary of each vertex group of $\mathcal G$.

Homeomorphism types of Floyd boundaries of infinite-ended groups

Abstract

Suppose is a finitely generated infinite group, and is a graph of groups decomposition of such that the edge groups are finite. This paper establishes that the topology of the Floyd boundary of is uniquely determined by the topology of the Floyd boundary of each vertex group of .
Paper Structure (20 sections, 31 theorems, 36 equations)

This paper contains 20 sections, 31 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coarse geometric notion
  4. Floyd completion of groups
  5. A tree of Cayley graphs corresponding to amalgamated free product
  6. A tree of Cayley graphs corresponding to HNN extension
  7. A Construction of Floyd boundaries of an amalgamated free product and HNN extension
  8. Boundaries of the stabilizers.
  9. Two topologies on $\overline{\Gamma}_f$
  10. Equivalence of topologies on $\overline{\Gamma}_f$
  11. HNN extension case
  12. Homeomorphism types of Floyd boundaries of free products
  13. Homeomorphism types of Floyd boundaries of amalgamated free product and HNN extension
  14. Amalgam case
  15. HNN extension Case
  16. ...and 5 more sections

Key Result

Lemma 2.3

If $\{w_n\}$ is a shortest sequence in $\Gamma_G$ then $\{w_n\}$ is a Cauchy sequence in $(\Gamma_G,d_f)$.

Theorems & Definitions (57)

  • Definition 2.1: Floyd boundary
  • Definition 2.2: Shortest sequence
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 47 more