Tilings of the Hyperbolic Space and Lipschitz Functions

Abstract

We use a special tiling for the hyperbolic $d$-space ${\mathbb{H}}^d$ for $d=2,3,4$ to construct an (almost) explicit isomorphism between the Lipschitz-free space $\mathcal{F}({\mathbb{H}}^d)$ and $\mathcal{F}(P)\oplus \mathcal{F}(\mathcal{N})$ where $P$ is a polytope in ${\mathbb{R}}^d$ and $\mathcal{N}$ a net in ${\mathbb{H}}^d$ coming from the tiling. This implies that the spaces $\mathcal{F}({\mathbb{H}}^d)$ and $\mathcal{F}({\mathbb{R}}^d)\oplus \mathcal{F}(\mathcal{M})$ are isomorphic for every net $\mathcal{M}$ in ${\mathbb{H}}^d$. In particular, we obtain that, for $d=2,3,4$, $\mathcal{F}({\mathbb{H}}^d)$ has a Schauder basis. Moreover, using a similar method, we also give an explicit isomorphism between $\mathrm{Lip}({\mathbb{H}}^d)$ and $\mathrm{Lip}({\mathbb{R}}^d)$.

22 pages, 1 figure