Bi-Lipschitz extension from boundaries of certain hyperbolic spaces

Tukia and Vaisala showed that every quasi-conformal map of ${\mathbb{R}}^n$ extends to a quasi-conformal self-map of ${\mathbb{R}}^{n+1}$. The restriction of the extended map to the upper half-space ${\mathbb{R}}^n\times {\mathbb{R}}^{+}$ is, in fact, bi-Lipschitz with respect to the hyperbolic metric. More generally, every homogeneous negatively curved manifold decomposes as $M=N⋊{\mathbb{R}}^{+}$ where $N$ is a nilpotent group with a metric on which ${\mathbb{R}}^{+}$ 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.