Vertical quasi-isometries and branched quasisymmetries

2019/11/28 by Lindquist, Jeff, Pankka, Pekka
#Complex Variables (math.CV) #FOS: Mathematics #Metric Geometry (math.MG) #Primary 30L10 #Secondary 30C65

paper · doi:10.48550/arxiv.1911.12680

Abstract

We introduce a class of mappings called vertical quasi-isometries and show that branched quasisymmetries X→ Y of Guo and Williams between compact, bounded turning metric doubling spaces admit natural vertically quasi-isometric extensions \widehat X→ \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 → \widehat Y between hyperbolic fillings induces a branched quasisymmetry X → Y.

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