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Quasi-isometry invariance of hyperbolicity in semimetric spaces, digraphs and semigroups

Matthias Hamann

Abstract

Gray and Kambites introduced a notion of hyperbolicity in the setting of semimetric spaces like digraphs or semigroups. We will prove that under a small additional geometric assumption their notion of hyperbolicity is preserved by quasi-isometries. Applied to semigroups, this will partially solve a problem of Gray and Kambites.

Quasi-isometry invariance of hyperbolicity in semimetric spaces, digraphs and semigroups

Abstract

Gray and Kambites introduced a notion of hyperbolicity in the setting of semimetric spaces like digraphs or semigroups. We will prove that under a small additional geometric assumption their notion of hyperbolicity is preserved by quasi-isometries. Applied to semigroups, this will partially solve a problem of Gray and Kambites.

Paper Structure

This paper contains 10 sections, 13 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semimetric spaces
  4. Digraphs
  5. Thin triangles
  6. Examples of hyperbolic semimetric spaces
  7. Divergence of geodesics
  8. Geodesic stability
  9. Quasi-isometries
  10. Hyperbolic semigroups

Key Result

Proposition 3.1

Let $X$ be a geodesic semimetric space and $\delta\geq 0$. If all transitive geodesic triangles are $\delta$-thin, then all geodesic triangles are $3\delta$-thin.

Theorems & Definitions (26)

  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 4.1
  • Proposition 4.2
  • ...and 16 more