Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A boundary for hyperbolic digraphs and semigroups

Matthias Hamann

Abstract

Based on a notion by Gray and Kambites of hyperbolicity in the setting of semimetric spaces like digraphs or semigroups, we will construct (under a small additional geometric assumption) a boundary based on quasi-geodesic rays and anti-rays that is preserved by quasi-isometries and, in the case of locally finite digraphs and right cancellative semigroups, refines their ends. Among other results, we show that it is possible to equip the space, if it is finitely based, together with its boundary with a pseudo-semimetric.

A boundary for hyperbolic digraphs and semigroups

Abstract

Based on a notion by Gray and Kambites of hyperbolicity in the setting of semimetric spaces like digraphs or semigroups, we will construct (under a small additional geometric assumption) a boundary based on quasi-geodesic rays and anti-rays that is preserved by quasi-isometries and, in the case of locally finite digraphs and right cancellative semigroups, refines their ends. Among other results, we show that it is possible to equip the space, if it is finitely based, together with its boundary with a pseudo-semimetric.
Paper Structure (13 sections, 35 theorems, 59 equations)

This paper contains 13 sections, 35 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semimetric spaces
  4. Digraphs
  5. Thin triangles
  6. Geodesic boundary
  7. Quasi-geodesic boundary
  8. Ends of digraphs
  9. Topologies for the quasi-geo-desic boundary
  10. A pseudo-semimetric for $X\cup\partial X$
  11. Properties of $X\cup\partial X$
  12. The size of the boundary
  13. Hyperbolic semigroups

Key Result

Proposition 2.1

Let $X$ be a semimetric space. Then the following hold.

Theorems & Definitions (62)

  • Proposition 2.1
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Proposition 3.4
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Proposition 4.3
  • ...and 52 more