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Visual maps between coarsely convex spaces

Yuuhei Ezawa, Tomohiro Fukaya

Abstract

The class of coarsely convex spaces is a coarse geometric analogue of the class of nonpositively curved Riemannian manifolds. It includes Gromov hyperbolic spaces, CAT(0) spaces, proper injective metric spaces and systolic complexes. It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces induce topological embeddings of their boundaries. Dydak and Virk studied maps between Gromov hyperbolic spaces which induce continuous maps between their boundaries. In this paper, we generalize their work to maps between coarsely convex spaces.

Visual maps between coarsely convex spaces

Abstract

The class of coarsely convex spaces is a coarse geometric analogue of the class of nonpositively curved Riemannian manifolds. It includes Gromov hyperbolic spaces, CAT(0) spaces, proper injective metric spaces and systolic complexes. It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces induce topological embeddings of their boundaries. Dydak and Virk studied maps between Gromov hyperbolic spaces which induce continuous maps between their boundaries. In this paper, we generalize their work to maps between coarsely convex spaces.

Paper Structure

This paper contains 25 sections, 150 equations.

Table of Contents

  1. Introduction
  2. Coarsely convex space
  3. Coarse map
  4. Coarse homotopy
  5. Coarsely convex space
  6. Ideal boundary
  7. Topology on the boundary
  8. Metric on the boundary
  9. Sequential boundary
  10. Visual map and radial map
  11. Visual map
  12. Maps compatible with systems of good quasi-geodesics
  13. Proof of Theorem \ref{['thm:fixed-pt']}
  14. Coarsely surjective
  15. Coarsely n-to-one map
  16. ...and 10 more sections

Theorems & Definitions (35)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 25 more