Sequences of asymptotically Möbius maps from ∂∞H² to ∂∞X converge after isometries to a map induced by an isometric embedding of H² into X when Isom(X) acts transitively on boundary triples.
Biswas, Circumcenter extension of Moebius maps to (-1) spaces, preprint, https://arxiv.org/pdf/1709,09110.pdf
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Asymptotically Moebius maps and rigidity for the hyperbolic plane
Sequences of asymptotically Möbius maps from ∂∞H² to ∂∞X converge after isometries to a map induced by an isometric embedding of H² into X when Isom(X) acts transitively on boundary triples.