# Bilipschitz Equivalence of Trees and Hyperbolic Fillings

## TL;DR

This paper demonstrates that under certain geometric conditions, quasi-isometries between specific metric spaces are close to bilipschitz maps, with applications to trees and hyperbolic fillings.

## Contribution

It establishes a link between quasi-isometries and bilipschitz maps for spaces satisfying linear isoperimetric inequalities, extending previous results.

## Key findings

- Quasi-isometries are within bounded distance of bilipschitz maps under given conditions.
- Application to regularly branching trees shows the practical relevance.
- Hyperbolic fillings of metric spaces also satisfy the bilipschitz equivalence.

## Abstract

We combine conditions found in [Wh] with results from [MPR] to show that quasi-isometries between uniformly discrete bounded geometry spaces that satisfy linear isoperimetric inequalities are within bounded distance to bilipschitz equivalences. We apply this result to regularly branching trees and hyperbolic fillings of metric spaces.