Vertical quasi-isometries and branched quasisymmetries

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 $\hat{X}\to \hat{Y}$ between hyperbolic fillings $\hat{X}$ and $\hat{Y}$ of $X$ and $Y$, respectively. We also give a converse for this result by showing that a finite multiplicity vertical quasi-isometry $\hat{X}\to \hat{Y}$ between hyperbolic fillings induces a branched quasisymmetry $X\to Y$.

50 pages, 1 figure

References in corpus (1)

• Geometric function theory: the art of pullback factorization

Cited by in corpus (1)

• Notes on quasiregular maps between Riemannian manifolds