Vertical quasi-isometries and branched quasisymmetries

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 $X \to Y$.