Real valued functions and metric spaces quasi-isometric to trees

TL;DR

This paper establishes conditions under which certain metric spaces are quasi-isometric to trees and provides criteria for spaces to be PQ-symmetric to ultrametric spaces, advancing understanding of their geometric structure.

Contribution

It introduces new conditions linking homology, properness, and bornologous maps to quasi-isometry with trees and ultrametric spaces.

Findings

Complete geodesic spaces with specific homology are quasi-isometric to trees.

Provides an intrinsic condition for metric spaces to be PQ-symmetric to ultrametric spaces.

Extends the theory of hyperbolic approximation in metric space analysis.

Abstract

We prove that if X is a complete geodesic metric space with uniformly generated first homology group and $f: X\to R$ is metrically proper on the connected components and bornologous, then X is quasi-isometric to a tree. Using this and adapting the definition of hyperbolic approximation we obtain an intrinsic sufficent condition for a metric space to be PQ-symmetric to an ultrametric space.