Tree approximation in quasi-trees

TL;DR

This paper explores the geometric structure of quasi-trees, demonstrating they are closely approximated by actual trees and improving existing tree approximation results for hyperbolic spaces.

Contribution

It introduces a new construction of trees approximating quasi-trees and enhances Gromov's tree approximation lemma for hyperbolic spaces.

Findings

Every quasi-tree is $(1,C)$-quasi-isometric to a simplicial tree.

Improved uniform approximation of hyperbolic spaces by trees.

Boundary of a quasi-tree is isometric to the boundary of its approximating tree.

Abstract

In this paper we investigate the geometric properties of quasi-trees, and prove some equivalent criteria. We give a general construction of a tree that approximates the ends of a geodesic space, and use this to prove that every quasi-tree is $(1,C)$-quasi-isometric to a simplicial tree. As a consequence, we show that Gromov's tree approximation lemma for hyperbolic spaces can be improved in the case of quasi-trees to give a uniform approximation for any set of points, independent of cardinality. From this we show that having uniform tree approximation for finite subsets is equivalent to being able to uniformly approximate the entire space by a tree. As another consequence, we note that the boundary of a quasi-tree is isometric to the boundary of its approximating tree under a certain choice of visual metric, and that this gives a natural extension of the standard metric on the boundary of a tree.