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 $φ: \partial_\infty S \rightarrow \partial_\infty X$ respects cross ratios, that means $\text{cr}_S( ξ_0,η_0,ξ_1,η_1)=\text{cr}_X( φ(ξ_0),φ(η_0),φ(ξ_1),φ(η_1))$ for every $ξ_0,η_0,ξ_1,η_1 \in \partial_\infty S$, then $φ$ 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 $φ_k: \partial_\infty \mathbb{H}^2 \rightarrow \partial_\infty X$ be a sequence of continuous maps which are asymptotically Moebius, that means $\lim_{k \to \infty} \text{cr}_X(φ_k(ξ_0),φ_k(η_0),φ_k(ξ_1),φ_k(η_1))=\text{cr}_{\mathbb{H}^2}( ξ_0,η_0,ξ_1,η_1)$ for every $ξ_0,η_0,ξ_1,η_1 \in \partial_\infty \mathbb{H}^2$. 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 $φ_\infty: \partial_\infty \mathbb{H}^2\rightarrow \partial_\infty X$ such that $\lim_{k \to \infty} g_kφ_k(ξ)=φ_\infty(ξ)$ for every $ξ\in \partial_\infty \mathbb{H}^2$ and $φ_\infty$ is induced by an isometric embedding of $\mathbb{H}^2$ into $X$.