Real valued functions and metric spaces quasi-isometric to trees

2011/03/30 by Martínez-Pérez, Álvaro
#53C23. Secondary: 20F65 #FOS: Mathematics #Geometric Topology (math.GT) #Primary: 54E35

paper · doi:10.48550/arxiv.1103.5908

Abstract

We prove that if X is a complete geodesic metric space with uniformly generated first homology group and f: X→ 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.

Related

• A metric between quasi-isometric trees
• Isomorphism of trees and isometry of ultrametric spaces
• Tree approximation in quasi-trees
• Quantitative correspondence between quasi-symmetric mappings on complete metric spaces and rough quasi-isometric mappings on their hyperbolic fillings
• Extremal properties and morphisms of finite ultrametric spaces and their representing trees