Vertical quasi-isometries and branched quasisymmetries

TL;DR

This paper introduces vertical quasi-isometries and demonstrates their relationship with branched quasisymmetries through extensions and converses in hyperbolic fillings of metric spaces.

Contribution

It establishes a new connection between branched quasisymmetries and vertical quasi-isometries via hyperbolic fillings, including extension and converse results.

Findings

Branched quasisymmetries admit natural vertical quasi-isometric extensions.

Vertical quasi-isometries induce branched quasisymmetries in the base spaces.

The results provide a new framework for understanding mappings between hyperbolic fillings.

Abstract

We introduce a class of mappings called vertical quasi-isometries and show that branched quasisymmetries $X\to Y$ of Guo and Williams between compact, bounded turning metric doubling spaces admit natural vertically quasi-isometric extensions $\widehat X\to \widehat Y$ between hyperbolic fillings $\widehat X$ and $\widehat Y$ of $X$ and $Y$, respectively. We also give a converse for this result by showing that a finite multiplicity vertical quasi-isometry $\widehat X \to \widehat Y$ between hyperbolic fillings induces a branched quasisymmetry $X \to Y$.