Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The metric Rips filtration, universal quasigeodesic cones, and hierarchically hyperbolic spaces

Robert Tang

TL;DR

The paper builds a cohesive, category‑theoretic framework for coarse geometry centered on the metric Rips filtration and its colimit, clarifying when a canonical large‑scale model of a space arises and proving colimit‑closure for the quasigeodesic class within the coarse Lipschitz setting. It then develops a universal quasigeodesic cone formalism via a Rips–Tuple recipe, enabling a uniform, non‑inductive construction of global cones from diagram data and applying this to hierarchically hyperbolic spaces (HHS). The HHS results show the total space is universal among uniformly controlled quasigeodesic cones, and the Rips–Tuple description identifies the total space with a Rips graph over the space of coarsely consistent tuples, abstracting the distance formula and hierarchy machinery. Beyond HHS, the framework yields two key advantages: (i) a systematic adjunction between quasigeodesic graphs and coarsely geodesic spaces via Rips colimits, and (ii) a local‑to‑global criterion for assembling factorwise retractions into a canonical global retraction, all under mild uniformity hypotheses. Overall, the work provides a flexible, diagrammatic toolkit for large‑scale geometry, enabling streamlined reconstructions and constructions across spaces endowed with pairwise constraints.

Abstract

We introduce a flexible, categorical framework for large-scale geometry that clarifies basic behaviour of the metric Rips filtration and streamlines some constructions in geometric group theory. The paper has two main parts. First, we develop the theory of the metric Rips filtration and its colimit in natural coarse categories: informally, we characterise when the Rips colimit produces a canonical large-scale model of a metric space and use this to prove that the quasigeodesic subcategory is closed under colimits in the coarsely Lipschitz category. We also establish adjointness properties of the Rips colimit and use them to characterise extremal metrics and universal morphisms from quasigeodesic sources. Second, we apply this machinery to characterise universal quasigeodesic cones via an explicit Rips-Tuple recipe. In the HHS setting this yields a concrete, canonical model of the total space: an HHS is quasi-isometric to a Rips graph of the space of coarsely consistent tuples in the product of its factor spaces. Moreover, we give a local-to-global criterion that promotes uniformly controlled, factorwise retractions to a canonical global hierarchical retraction. Because the approach is based on universal properties and uniformly controlled coarse data rather than inductive constructions, distance formulae, or hierarchy paths, it applies equally well to arbitrary families of metric spaces equipped with pairwise constraints.

The metric Rips filtration, universal quasigeodesic cones, and hierarchically hyperbolic spaces

TL;DR

The paper builds a cohesive, category‑theoretic framework for coarse geometry centered on the metric Rips filtration and its colimit, clarifying when a canonical large‑scale model of a space arises and proving colimit‑closure for the quasigeodesic class within the coarse Lipschitz setting. It then develops a universal quasigeodesic cone formalism via a Rips–Tuple recipe, enabling a uniform, non‑inductive construction of global cones from diagram data and applying this to hierarchically hyperbolic spaces (HHS). The HHS results show the total space is universal among uniformly controlled quasigeodesic cones, and the Rips–Tuple description identifies the total space with a Rips graph over the space of coarsely consistent tuples, abstracting the distance formula and hierarchy machinery. Beyond HHS, the framework yields two key advantages: (i) a systematic adjunction between quasigeodesic graphs and coarsely geodesic spaces via Rips colimits, and (ii) a local‑to‑global criterion for assembling factorwise retractions into a canonical global retraction, all under mild uniformity hypotheses. Overall, the work provides a flexible, diagrammatic toolkit for large‑scale geometry, enabling streamlined reconstructions and constructions across spaces endowed with pairwise constraints.

Abstract

We introduce a flexible, categorical framework for large-scale geometry that clarifies basic behaviour of the metric Rips filtration and streamlines some constructions in geometric group theory. The paper has two main parts. First, we develop the theory of the metric Rips filtration and its colimit in natural coarse categories: informally, we characterise when the Rips colimit produces a canonical large-scale model of a metric space and use this to prove that the quasigeodesic subcategory is closed under colimits in the coarsely Lipschitz category. We also establish adjointness properties of the Rips colimit and use them to characterise extremal metrics and universal morphisms from quasigeodesic sources. Second, we apply this machinery to characterise universal quasigeodesic cones via an explicit Rips-Tuple recipe. In the HHS setting this yields a concrete, canonical model of the total space: an HHS is quasi-isometric to a Rips graph of the space of coarsely consistent tuples in the product of its factor spaces. Moreover, we give a local-to-global criterion that promotes uniformly controlled, factorwise retractions to a canonical global hierarchical retraction. Because the approach is based on universal properties and uniformly controlled coarse data rather than inductive constructions, distance formulae, or hierarchy paths, it applies equally well to arbitrary families of metric spaces equipped with pairwise constraints.

Paper Structure

This paper contains 18 sections, 86 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Coarse geometry
  3. The metric Rips filtration and its colimit
  4. The Rips filtration and controlled maps
  5. Rips colimit in the metric coarse category
  6. $\mathsf{C}$--geodesic spaces
  7. Colimits of $\mathsf{C}$--geodesic spaces
  8. Rips colimit under dominated controls
  9. Adjointness and consequences
  10. Universal morphisms
  11. Extremality of the stable Rips colimit
  12. Universal $\mathsf{C}$--geodesic cones
  13. Equalisers
  14. Uniformly controlled cones
  15. Interactions with retractions
  16. ...and 3 more sections

Key Result

Theorem 1.1

Every metric space is canonically isomorphic to its Rips colimit in $\overline{\bm{\mathnormal{\mathsf{Coarse}}}\xspace\xspace}$.

Theorems & Definitions (117)

  • Theorem 1.1: $\overline{\bm{\mathnormal{\mathsf{Coarse}}}\xspace\xspace}$--Rips colimit
  • Theorem 1.2: $\overline{\bm{\mathnormal{\mathsf{CLip}}}\xspace\xspace}$--Rips colimit
  • Theorem 1.3: Coclosed subcategory
  • Corollary 1.4: Finite cocompleteness
  • Remark 1.5
  • Theorem 1.6: Adjointness
  • Theorem 1.7: Universal morphisms
  • Theorem 1.8: Extremal metrics
  • Theorem 1.9: Universal quasigeodesic cones
  • Theorem 1.10: Rips--Tuple recipe
  • ...and 107 more