[Paper Review] Hyperbolicity, CAT(-1)-spaces and the Ptolemy Inequality

Thomas Foertsch, Viktor Schroeder|ArXiv.org|May 16, 2006

Geometric and Algebraic Topology6 references4 citations

TL;DR

This paper establishes a necessary condition for a metric space to arise as the visual boundary of a CAT(-1) space: the Ptolemy inequality, which generalizes classical Ptolemy's theorem to hyperbolic geometry. Using a four-point inequality derived from Bourdon's work, the authors show that the boundary of any CAT(-1) space satisfies this Möbius-invariant inequality, and equality holds precisely when the four points form an ideal quadrilateral in the hyperbolic plane. This provides a geometric obstruction to a space being a visual boundary of a CAT(-1) space.

ABSTRACT

Using a four points inequality for the boundary of CAT(-1)-spaces, we study the relation between Gromov hyperbolic spaces and CAT(-1)-spaces.

Motivation & Objective

• To identify a necessary geometric condition for a metric space to arise as the visual boundary of a CAT(-1) space.
• To analyze the relationship between Gromov hyperbolic spaces and CAT(-1) spaces using boundary geometry.
• To provide a characterization of when a Gromov hyperbolic space is roughly isometric to a CAT(-1) space via normalization using the asymptotic upper curvature bound.
• To demonstrate that the Ptolemy inequality is invariant under Möbius transformations and thus well-suited for boundary geometry.

Proposed method

• Derives a four-point inequality from Bourdon's result on the boundary of CAT(-1) spaces.
• Introduces the Ptolemy inequality as a Möbius-invariant condition on the visual boundary of CAT(-1) spaces.
• Uses the critical metric normalization (K_u(X) = -1) to study rough isometries between visual Gromov hyperbolic spaces and CAT(-1) spaces.
• Applies the Assouad embedding theorem and snowflake maps to construct embeddings into Hilbert spaces and then into boundaries of CAT(-1) spaces.
• Constructs a 1/2-snowflake map from the unit ball in ℓ¹ into the boundary of an infinite-dimensional CAT(-1) space using stereographic projection and bi-Lipschitz embeddings.
• Leverages Lang-Schlichenmaier and Alexander-Bishop results to show that finite Assouad-Nagata dimension implies rough isometry after small scaling.

Experimental results

Research questions

• RQ1 Which metric spaces can appear as the visual boundary of a CAT(-1) space?
• RQ2 What necessary geometric condition must a boundary space satisfy to be realizable as the boundary of a CAT(-1) space?
• RQ3 Under what conditions is a visual Gromov hyperbolic space roughly isometric to a CAT(-1) space?
• RQ4 How does the Ptolemy inequality relate to the geometry of ideal quadrilaterals in hyperbolic space?
• RQ5 Can snowflake maps from spaces of finite Assouad-Nagata dimension be used to construct embeddings into CAT(-1) boundaries?

Key findings

• The boundary of any CAT(-1) space satisfies the Ptolemy inequality: |y₁y₃||y₂y₄| ≤ |y₁y₂||y₃y₄| + |y₂y₃||y₄y₁| for all four points on the boundary.
• Equality in the Ptolemy inequality holds if and only if the convex hull of the four points is isometric to an ideal quadrilateral in the hyperbolic plane with the two diagonals being y₁y₃ and y₂y₄.
• The Ptolemy inequality is invariant under Möbius transformations, making it a natural condition for boundary geometry of CAT(-1) spaces.
• For any nontreelike visual Gromov hyperbolic space with doubling boundary, scaling the critical metric by any λ < 1 yields a space roughly isometric to a CAT(-1) space.
• If the boundary has finite Assouad-Nagata dimension, then some small scaling of the critical metric yields a rough isometry to a CAT(-1) space.
• A 1/2-snowflake map exists from the unit ball in ℓ¹ into the boundary of an infinite-dimensional CAT(-1) space, constructed via stereographic projection and bi-Lipschitz embedding into Hilbert space.