Circumcenter extension of Moebius maps to CAT(-1) spaces

2017/09/26 by Biswas, Kingshook
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1709.09110

Abstract

Given a Moebius homeomorphism f : ∂ X → ∂ Y between boundaries of proper, geodesically complete CAT(-1) spaces X,Y, we describe an extension f : X → Y of f, called the circumcenter map of f, which is constructed using circumcenters of expanding sets. The extension f is shown to coincide with the (1, log 2)-quasi-isometric extension constructed in [biswas3], and is locally 1/2-Holder continuous. When X,Y are complete, simply connected manifolds with sectional curvatures K satisfying -b² ≤ K ≤ -1 for some b ≥ 1 then the extension f : X → Y is a (1, (1 - (1)/(b))log 2)-quasi-isometry. Circumcenter extension of Moebius maps is natural with respect to composition with isometries.

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