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Asymptotic geometric regularity of CAT(0) spaces

Koichi Nagano

Abstract

We prove that if an n-dimensional geodesically complete CAT(0) space has Tits boundary sufficiently close to the (n-1)-dimensional standard unit sphere, then it is bi-Lipschiz homeomorphic to the n-dimensional Euclidean space. As an application, we conclude that if an (n-1)-dimensional geodesically complete CAT(1) space is sufficiently close to the (n-1)-dimensional standard unit sphere, then they are bi-Lipschiz homeomorphic to each other.

Asymptotic geometric regularity of CAT(0) spaces

Abstract

We prove that if an n-dimensional geodesically complete CAT(0) space has Tits boundary sufficiently close to the (n-1)-dimensional standard unit sphere, then it is bi-Lipschiz homeomorphic to the n-dimensional Euclidean space. As an application, we conclude that if an (n-1)-dimensional geodesically complete CAT(1) space is sufficiently close to the (n-1)-dimensional standard unit sphere, then they are bi-Lipschiz homeomorphic to each other.
Paper Structure (25 sections, 24 theorems, 49 equations)

This paper contains 25 sections, 24 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Main theorems
  3. Organization
  4. Preliminaries
  5. Metric spaces
  6. Maps between metric spaces
  7. Geodesic metric spaces
  8. Gromov--Hausdorff topology
  9. CAT$\boldsymbol{(\kappa)}$ spaces
  10. Ideal boundaries of $\operatorname{CAT}(0)$ spaces
  11. Geodesically complete CAT$\boldsymbol{(\kappa)}$ spaces
  12. Dimensions of CAT$\boldsymbol{(\kappa)}$ spaces
  13. Almost spherical suspension structure
  14. Multi-fold suspenders in CAT(1) spaces
  15. Relaxed multi-fold suspenders in CAT(1) spaces
  16. ...and 10 more sections

Key Result

Theorem 1.1

For every $\epsilon \in (0,\infty)$, and for every $n \in \operatorname{\mathbb{N}}$, there exists $\delta \in (0,\infty)$ such that if a separable, geodesically complete $\operatorname{CAT}(0)$ space $X$ of $\dim X \le n$ satisfies $d_{\operatorname{\mathrm{GH}}} \left( \partial_{\mathrm{T}}X, \ope

Theorems & Definitions (44)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Theorem 2.1
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 34 more