[Paper Review] Uniformly continuous maps between ends of R-trees

Álvaro Martinez Pérez, Manuel A. Morón|ArXiv.org|Apr 24, 2007

Topological and Geometric Data Analysis14 citations

TL;DR

This paper establishes a categorical equivalence between complete bounded ultrametric spaces with uniformly continuous maps and geodesically complete rooted ℝ-trees with metrically proper non-expansive maps, providing an explicit construction of such maps via a modulus of continuity derived from Borsuk's method. The key contribution is a uniform correspondence between uniform structures on end spaces and coarse-geometric properties of ℝ-trees.

ABSTRACT

There is a well-known correspondence between infinite trees and ultrametric spaces which can be interpreted as an equivalence of categories and comes from considering the end space of the tree. In this equivalence, uniformly continuous maps between the end spaces are translated to some classes of coarse maps (or even classes of metrically proper lipschitz maps) between the trees.

Motivation & Objective

• To characterize the uniform type of end spaces of geodesically complete rooted ℝ-trees using coarse geometry.
• To extend Bruce Hughes's categorical equivalences beyond isomorphisms to include all uniformly continuous maps.
• To construct an explicit non-expansive map between ℝ-trees that induces a given uniformly continuous map between their end spaces.
• To recover the classical Freudenthal end space classification via proper homotopy types in the context of ℝ-trees.

Proposed method

• Use Borsuk's procedure to associate a modulus of continuity to a uniformly continuous map between end spaces.
• Construct a non-expansive map between ℝ-trees by lifting the modulus of continuity to the tree structure.
• Define metrically proper continuous maps between ℝ-trees that induce uniformly continuous maps on their end spaces.
• Establish equivalence between categories: complete bounded ultrametric spaces with uniformly continuous maps and geodesically complete rooted ℝ-trees with metrically proper non-expansive maps.
• Use the Hopf-Rinow theorem and properness to ensure completeness and local compactness of pruned subtrees.
• Apply retraction maps to relate locally finite simplicial trees to their pruned, complete, geodesically complete subtrees.

Experimental results

Research questions

• RQ1 How can uniformly continuous maps between end spaces of ℝ-trees be systematically lifted to maps between the trees themselves?
• RQ2 What geometric properties of ℝ-trees correspond to uniform equivalence types of their end spaces?
• RQ3 Can the categorical equivalence between ultrametric spaces and ℝ-trees be extended beyond isomorphisms to include all uniformly continuous maps?
• RQ4 What role does the modulus of continuity play in translating uniform structures on ultrametric spaces to coarse-geometric structures on ℝ-trees?
• RQ5 How do proper homotopy types of locally finite trees relate to the topological type of their Freudenthal end spaces?

Key findings

• The category of complete bounded ultrametric spaces with uniformly continuous maps is isomorphic to the category of geodesically complete rooted ℝ-trees with metrically proper continuous maps.
• An explicit formula constructs a non-expansive map between ℝ-trees that induces any given uniformly continuous map between their end spaces.
• Proper homotopy equivalence between locally finite simplicial trees corresponds exactly to homeomorphism of their Freudenthal end spaces.
• The end space of a geodesically complete rooted ℝ-tree is uniformly homeomorphic to the end space of its pruned, complete, geodesically complete subtree.
• Non-rooted isometries between ℝ-trees induce bi-Lipschitz maps between their end spaces with distortion bounded by exponential of the translation distance.
• Metrically proper non-expansive maps between ℝ-trees induce uniformly continuous maps between end spaces, and vice versa, establishing a full categorical equivalence.