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

Abstract

Given a Moebius homeomorphism $f:\partial X\to \partial Y$ between boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, we describe an extension $\hat{f}:X\to Y$ of $f$, called the circumcenter map of $f$, which is constructed using circumcenters of expanding sets. The extension $\hat{f}$ 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 $-b^2\le K\le -1$ for some $b\ge 1$ then the extension $\hat{f}:X\to Y$ is a $(1,(1-\frac{1}{b})\log 2)$-quasi-isometry. Circumcenter extension of Moebius maps is natural with respect to composition with isometries.

arXiv commentAdded local Holder continuity of circumcenter extension. Minor corrections made. arXiv admin note: text overlap with arXiv:1203.6212