Bi-Lipschitz extension from boundaries of certain hyperbolic spaces

TL;DR

This paper extends quasi-symmetries of certain hyperbolic space boundaries to bi-Lipschitz maps of the entire space, broadening understanding of geometric extensions in negatively curved manifolds.

Contribution

It generalizes bi-Lipschitz extension results to a wider class of hyperbolic manifolds with nilpotent group boundaries, beyond classical rank one symmetric spaces.

Findings

Quasi-symmetries of boundary spaces extend to bi-Lipschitz maps of the entire manifold.

Applicable to non-compact rank one symmetric spaces and other negatively curved manifolds.

Results hold even when curvature is not strictly negative.

Abstract

Tukia and Vaisala showed that every quasi-conformal map of $\R^n$ extends to a quasi-conformal self-map of $\R^{n+1}$. The restriction of the extended map to the upper half-space $\R^n \times \R^+$ is, in fact, bi-Lipschitz with respect to the hyperbolic metric. More generally, every homogeneous negatively curved manifold decomposes as $M = N \rtimes \R^+$ where $N$ is a nilpotent group with a metric on which $\R^+$ acts by dilations. We show that under some assumptions on $N$, every quasi-symmetry of $N$ extends to a bi-Lipschitz map of $M$. The result applies to a wide class of manifolds $M$ including non-compact rank one symmetric spaces and certain manifolds that do not admit co-compact group actions. Although $M$ must be Gromov hyperbolic, its curvature need not be strictly negative.