[Paper Review] Uniformizing Gromov hyperbolic spaces and Busemann functions

Qingshan Zhou|arXiv (Cornell University)|Aug 4, 2020

Geometric and Algebraic Topology56 references4 citations

TL;DR

This paper introduces a novel metric density using Busemann functions to uniformize unbounded Gromov hyperbolic spaces, establishing a one-to-one correspondence between quasi-isometry classes of proper geodesic, roughly starlike hyperbolic spaces and quasi-similarity classes of unbounded locally compact uniform spaces. The method yields new proofs of classical results and extends Teichmüller’s displacement theorem and quasisymmetry theorems in metric settings.

ABSTRACT

By introducing a new metric density via Busemann function, we establish an unbounded uniformizing Gromov hyperbolic spaces procedure which is an analogue of a recent work of Bonk, Heinonen and Koskela in \cite{BHK}. Then we show that there is a one-to-one correspondence between the quasi-isometry classes of proper geodesic Gromov hyperbolic spaces that are roughly starlike with respect to the points at the boundaries of infinity and the quasi-similarity classes of unbounded locally compact uniform spaces. As applications, we establish Teichmuller's displacement theorem for roughly quasi-isometry in Gromov hyperbolic spaces, and explain the connections to the bilipschitz extensions of certain Gromov hyperbolic spaces. By using our uniformizing procedure, we also provide a new proof for Vaisala-Heinonen-Nakki's Theorem in the setting of metric spaces. Moreover, we obtain the quasisymmetry from local to global on uniform metric spaces.

Motivation & Objective

• To develop a uniformizing procedure for unbounded Gromov hyperbolic spaces using a new metric density derived from Busemann functions.
• To establish a one-to-one correspondence between quasi-isometry classes of roughly starlike proper geodesic Gromov hyperbolic spaces and quasi-similarity classes of unbounded locally compact uniform spaces.
• To apply the uniformizing method to prove Teichmüller’s displacement theorem for roughly quasi-isometries in Gromov hyperbolic spaces.
• To clarify connections between bilipschitz extensions and the geometric structure of Gromov hyperbolic spaces.
• To provide a new proof of Väisälä–Heinonen–Näkki’s theorem in the context of general metric spaces using the proposed uniformization.

Proposed method

• Defining a new metric density based on the Busemann function associated with geodesic rays in Gromov hyperbolic spaces.
• Constructing a conformal deformation of the original metric using this density to produce a uniform metric space.
• Proving that the resulting space is locally compact and uniformly perfect, satisfying the axioms of a uniform space.
• Establishing a bi-Lipschitz equivalence between the uniformized space and the original hyperbolic space under controlled distortion.
• Using the correspondence between geometric classes to transfer properties such as quasisymmetry and quasi-isometry invariance.
• Applying the uniformizing map to recover known results, including Väisälä–Heinonen–Näkki’s theorem, via a new geometric argument.

Experimental results

Research questions

• RQ1 How can Busemann functions be used to define a metric density that uniformizes unbounded Gromov hyperbolic spaces?
• RQ2 What is the precise correspondence between quasi-isometry classes of roughly starlike proper geodesic Gromov hyperbolic spaces and quasi-similarity classes of unbounded uniform spaces?
• RQ3 Can the uniformizing procedure be used to reprove classical results such as Teichmüller’s displacement theorem in the setting of roughly quasi-isometries?
• RQ4 What are the implications of this uniformization for bilipschitz extension properties in Gromov hyperbolic spaces?
• RQ5 How does the method enable the global extension of local quasisymmetries to global ones on uniform metric spaces?

Key findings

• A new metric density derived from Busemann functions successfully uniformizes unbounded Gromov hyperbolic spaces, transforming them into locally compact uniform spaces.
• There exists a one-to-one correspondence between quasi-isometry classes of proper geodesic, roughly starlike Gromov hyperbolic spaces and quasi-similarity classes of unbounded locally compact uniform spaces.
• The method provides a new proof of Väisälä–Heinonen–Näkki’s theorem in the general metric space setting, independent of previous analytic techniques.
• Teichmüller’s displacement theorem is extended to the setting of roughly quasi-isometries in Gromov hyperbolic spaces via the uniformization framework.
• The paper establishes a global quasisymmetry result by showing that local quasisymmetries on uniform metric spaces extend to global quasisymmetries using the uniformizing map.
• The uniformizing procedure reveals structural connections between bilipschitz extensions and the geometry of Gromov hyperbolic spaces, particularly in the context of boundary behavior.