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Coarse medians and universal quasigeodesic cones

Robert Tang

Abstract

We show that any universal quasigeodesic cone of uniformly coarse median spaces admits a canonical coarse median structure. As an application, we recover a result of Bowditch which states that any hierarchically hyperbolic space admits a coarse median structure compatible with the projections to the hyperbolic factor spaces.

Coarse medians and universal quasigeodesic cones

Abstract

We show that any universal quasigeodesic cone of uniformly coarse median spaces admits a canonical coarse median structure. As an application, we recover a result of Bowditch which states that any hierarchically hyperbolic space admits a coarse median structure compatible with the projections to the hyperbolic factor spaces.
Paper Structure (8 sections, 19 theorems, 13 equations)

This paper contains 8 sections, 19 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Coarse geometry
  3. Uniformly controlled cones
  4. Universal quasigeodesic cones
  5. Coarse median spaces
  6. Coarse median cones
  7. Quasigeodesic coarse median cones
  8. Bounded coarse interval image

Key Result

Theorem 1.1

Let $D$ be a uniformly controlled diagram of uniformly coarse median spaces $(X_i, \mu_i)$ whose maps are uniformly coarse median preserving. Then any universal quasigeodesic uniformly controlled cone $X$ over $D$ admits a canonical (up to closeness) coarse median $\mu$ such that all legs $(X,\mu) \

Theorems & Definitions (20)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 2.2: Stable tuplespaces Tang-rips
  • Lemma 2.3: Tang-rips
  • Proposition 2.4: Tang-rips
  • Theorem 2.5: Rips--tuple recipe Tang-rips
  • Definition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • ...and 10 more