Quantitative correspondence between quasi-symmetric mappings on complete metric spaces and rough quasi-isometric mappings on their hyperbolic fillings

TL;DR

This paper establishes a precise quantitative link between power quasi-symmetric mappings on complete metric spaces and rough quasi-isometries on their hyperbolic fillings, generalizing previous results from trees to hyperbolic spaces.

Contribution

It demonstrates that the exponents in quasi-symmetric mappings match the coefficients in rough quasi-isometries, extending prior work to hyperbolic fillings.

Findings

Exponents in quasi-symmetric mappings coincide with coefficients in quasi-isometries.

The correspondence is sharp and consistent.

Generalizes results from rooted trees to hyperbolic fillings.

Abstract

In this paper, we establish a quantitative correspondence between power quasi-symmetric mappings on complete metric spaces and rough quasi-isometric mappings on their hyperbolic fillings. In particular, we prove that the exponents in the power quasi-symmetric mappings coincide with the coefficients in the rough quasi-isometric mappings. This shows that the obtained correspondence is both sharp and consistent. In this way, we generalize the corresponding result by Bj\"orn, Bj\"orn, Gill, and Shanmugalingam (J. Reine Angew. Math., 2017) from the setting of rooted trees to that of hyperbolic fillings.