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Extending and improving conical bicombings

Giuliano Basso

Abstract

We study metric spaces that admit a conical bicombing and thus obey a weak form of non-positive curvature. Prime examples of such spaces are injective metric spaces. In this article we give a complete characterization of complete metric spaces admitting a conical bicombing by showing that every such space is isometric to a closed $σ$-convex subset of some injective metric space. In addition, we show that every proper metric space that admits a conical bicombing also admits a consistent bicombing that satisfies certain convexity conditions. This can be seen as a strong indication that a question from Descombes and Lang about improving conical bicombings might have a positive answer. As an application, we prove that any group acting geometrically on a proper metric space with a conical bicombing admits a $\mathcal{Z}$-structure.

Extending and improving conical bicombings

Abstract

We study metric spaces that admit a conical bicombing and thus obey a weak form of non-positive curvature. Prime examples of such spaces are injective metric spaces. In this article we give a complete characterization of complete metric spaces admitting a conical bicombing by showing that every such space is isometric to a closed -convex subset of some injective metric space. In addition, we show that every proper metric space that admits a conical bicombing also admits a consistent bicombing that satisfies certain convexity conditions. This can be seen as a strong indication that a question from Descombes and Lang about improving conical bicombings might have a positive answer. As an application, we prove that any group acting geometrically on a proper metric space with a conical bicombing admits a -structure.

Paper Structure

This paper contains 21 sections, 28 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Extending conical bicombings
  3. Improving conical bicombings
  4. Acknowledgements
  5. $1$-Wasserstein distances and barycentric metric spaces
  6. The $1$-Wasserstein distance
  7. Barycentric metric spaces
  8. Extending conical bicombings
  9. Consequences of the conical inequality
  10. Partially defined barycenter maps
  11. Extensions to the injective hull
  12. Doss expectation
  13. Conical bicombings as fixed points
  14. Conical bicombings on $\mathop{\mathrm{CB}}\nolimits(X)$
  15. A fixed point result and its applications
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $X$ be a complete metric space. Then the following statements are equivalent:

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 45 more