Asymptotically Moebius maps and rigidity for the hyperbolic plane

Abstract

Let $S$ be a rank-one symmetric space of non-compact type and let $X$ be a $\text{CAT}(-1)$ space. A well-known result by Bourdon states that if a topological embedding $\phi :{\partial }_{\infty }S\to {\partial }_{\infty }X$ respects cross ratios, that means ${\text{cr}}_S({\xi }_0,{\eta }_0,{\xi }_1,{\eta }_1)={\text{cr}}_X(\phi ({\xi }_0),\phi ({\eta }_0),\phi ({\xi }_1),\phi ({\eta }_1))$ for every ${\xi }_0,{\eta }_0,{\xi }_1,{\eta }_1\in {\partial }_{\infty }S$, then $\phi$ is induced by an isometric embedding of $S$ into $X$. We generalize this result when $S={\mathbb{H}}^2$ is the real hyperbolic plane as it follows. Let ${\phi }_k:{\partial }_{\infty }{\mathbb{H}}^2\to {\partial }_{\infty }X$ be a sequence of continuous maps which are asymptotically Moebius, that means ${\lim }_{k\to \infty }{\text{cr}}_X({\phi }_k({\xi }_0),{\phi }_k({\eta }_0),{\phi }_k({\xi }_1),{\phi }_k({\eta }_1))={\text{cr}}_{{\mathbb{H}}^2}({\xi }_0,{\eta }_0,{\xi }_1,{\eta }_1)$ for every ${\xi }_0,{\eta }_0,{\xi }_1,{\eta }_1\in {\partial }_{\infty }{\mathbb{H}}^2$. Assume that the isometry group $\text{Isom}(X)$ acts transitively on triples of distinct points of ${\partial }_{\infty }X$. Then there must exists a sequence $(g_k{)}_{k\in \mathbb{N}}$, $g_k\in \text{Isom}(X)$ and a map ${\phi }_{\infty }:{\partial }_{\infty }{\mathbb{H}}^2\to {\partial }_{\infty }X$ such that ${\lim }_{k\to \infty }{g}_k{\phi }_k(\xi )={\phi }_{\infty }(\xi )$ for every $\xi \in {\partial }_{\infty }{\mathbb{H}}^2$ and ${\phi }_{\infty }$ is induced by an isometric embedding of ${\mathbb{H}}^2$ into $X$.

10 pages