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1 Introduction 1.1 Background and statement of results An (L, C) quasi-isometry is a map Φ : X −→ X ′ between metric spaces such that for all x 1 , x 2 ∈ X we have L −1 d(x 1 , x 2) − C ≤ d(Φ(x 1), Φ(x 2)) ≤ Ld(x 1 , x 2) + C (1) and d(x ′ , Im(Φ)) < C (2) for all x ′ ∈ X ′. Quasi-isometries occur naturally in the study of the geometry of discrete groups(More)
We call a complement of a union of at least three disjoint (round) open balls in the unit sphere S n a Schottky set. We prove that every quasisymmetric homeomorphism of a Schottky set of spherical measure zero to another Schottky set is the restriction of a Möbius transformation on S n. In the other direction we show that every Schottky set in S 2 of(More)