2019/06/25 by Alessio Savini, Savini, Alessio
Mathematics · #53C24 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:53C24
paper · pdf · doi:10.48550/arxiv.1906.10563
10 pages
arxiv created 2019/06/25 · arxiv updated 2019/06/26
Let S be a rank-one symmetric space of non-compact type and let X be a CAT(-1) space. A well-known result by Bourdon states that if a topological embedding φ: ∂_∞ S → ∂_∞ X respects cross ratios, that means crS( ξ0,η0,ξ1,η1)=crX( φ(ξ0),φ(η0),φ(ξ1),φ(η1)) for every ξ0,η0,ξ1,η1 ∈ ∂_∞ S, then φ is induced by an isometric embedding of S into X. We generalize this result when S=ℍ2 is the real hyperbolic plane as it follows. Let φk: ∂_∞ ℍ2 → ∂_∞ X be a sequence of continuous maps which are asymptotically Moebius, that means limk → ∞ crX(φk(ξ0),φk(η0),φk(ξ1),φk(η1))=crℍ2( ξ0,η0,ξ1,η1) for every ξ0,η0,ξ1,η1 ∈ ∂_∞ ℍ2. Assume that the isometry group Isom(X) acts transitively on triples of distinct points of ∂_∞ X. Then there must exists a sequence (gk)k ∈ ℕ, gk ∈ Isom(X) and a map φ_∞: ∂_∞ ℍ2→ ∂_∞ X such that limk → ∞ gkφk(ξ)=φ_∞(ξ) for every ξ∈ ∂_∞ ℍ2 and φ_∞ is induced by an isometric embedding of ℍ2 into X.