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 $\varphi: \partial_\infty S \rightarrow \partial_\infty X$ respects cross ratios, that means $\text{cr}_S( \xi_0,\eta_0,\xi_1,\eta_1)=\text{cr}_X( \varphi(\xi_0),\varphi(\eta_0),\varphi(\xi_1),\varphi(\eta_1))$ for every $\xi_0,\eta_0,\xi_1,\eta_1 \in \partial_\infty S$, then $\varphi$ 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 $\varphi_k: \partial_\infty \mathbb{H}^2 \rightarrow \partial_\infty X$ be a sequence of continuous maps which are asymptotically Moebius, that means $\lim_{k \to \infty} \text{cr}_X(\varphi_k(\xi_0),\varphi_k(\eta_0),\varphi_k(\xi_1),\varphi_k(\eta_1))=\text{cr}_{\mathbb{H}^2}( \xi_0,\eta_0,\xi_1,\eta_1)$ 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 $\varphi_\infty: \partial_\infty \mathbb{H}^2\rightarrow \partial_\infty X$ such that $\lim_{k \to \infty} g_k\varphi_k(\xi)=\varphi_\infty(\xi)$ for every $\xi \in \partial_\infty \mathbb{H}^2$ and $\varphi_\infty$ is induced by an isometric embedding of $\mathbb{H}^2$ into $X$.