# Quasiflats in Hadamard spaces

Lang, U; Schroeder, Viktor (1997). Quasiflats in Hadamard spaces. Annales Scientifiques de l'Ecole Normale Superieure, 30(3):339-352.

## Abstract

Let X be a simply connected, complete geodesic metric space which is nonpositively curved in the sense of Alexandrov. We assume that X contains a k-flat F of maximal dimension and consider quasiisometric embeddings f : ℝk → X whose distance function from F satisfies a certain asymptotic growth condition. We prove that if X is locally compact and cocompact, then the Hausdorff distance between f(ℝk) and F is uniformly bounded. This generalizes a well-known lemma of Mostow on quasiflats in symmetric spaces of noncompact type.

## Abstract

Let X be a simply connected, complete geodesic metric space which is nonpositively curved in the sense of Alexandrov. We assume that X contains a k-flat F of maximal dimension and consider quasiisometric embeddings f : ℝk → X whose distance function from F satisfies a certain asymptotic growth condition. We prove that if X is locally compact and cocompact, then the Hausdorff distance between f(ℝk) and F is uniformly bounded. This generalizes a well-known lemma of Mostow on quasiflats in symmetric spaces of noncompact type.

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