Quasiflats in Hadamard spaces. (English) Zbl 0876.53050

Let \(X\) be a Hadamard space in the sense of Alexandrov, that is a simply connected complete geodesic metric space which is non-positively curved in the sense of Alexandrov [W. Ballmann, ‘Lectures on spaces of nonpositive curvature’ (DMV Seminar 25, Birkhäuser, Basel) (1995; Zbl 0834.53003); W. Ballmann, M. Gromov and V. Schroeder, ‘Manifolds of nonpositive curvature’ (Progress in Mathematics 61, Birkhäuser, Basel) (1985; Zbl 0591.53001)]. Suppose \(X\) contains a \(k\)-flat \(F\) of maximal dimension and consider quasiflats (i.e., quasi-isometric embeddings) \(f:\mathbb{R}^k\to X\) whose distance function from \(F\) satisfies a certain asymptotic growth condition. A known lemma of Mostow on quasiflats in symmetric spaces of non-compact type [G. D. Mostow, ‘Strong rigidity of locally symmetric spaces’ (Ann. Math. Studies 78, Princeton Univ. Pr.) (1973; Zbl 0265.53039)] is generalized here by the following result: the Hausdorff distance between \(f(\mathbb{R}^k)\) and \(F\) is uniformly bounded if \(X\) is locally compact and cocompact.
Reviewer: C.-L.Bejan (Iaşi)


53C70 Direct methods (\(G\)-spaces of Busemann, etc.)
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[1] W. BALLMANN , Lectures on spaces of nonpositive curvature , DMV Seminar Band, No. 25, Basel : Birkhäuser 1995 . MR 97a:53053 | Zbl 0834.53003 · Zbl 0834.53003
[2] W. BALLMANN , M. GROMOV and V. SCHROEDER : Manifolds of nonpositive curvature , Basel, Birkhäuser 1985 . MR 87h:53050 | Zbl 0591.53001 · Zbl 0591.53001
[3] M. R. BRIDSON , On the existence of flat planes in spaces of non-positive curvature (Proc. Amer. Math. Soc., Vol. 123, 1995 , pp. 223-235). MR 95d:53048 | Zbl 0840.53033 · Zbl 0840.53033
[4] M. R. BRIDSON and A. HAEFLIGER , Metric spaces of non-positive curvature , in preparation. · Zbl 0988.53001
[5] H. FEDERER , Geometric measure theory , Berlin, Springer 1969 . MR 41 #1976 | Zbl 0176.00801 · Zbl 0176.00801
[6] M. KAPOVICH and B. LEEB , Quasi-isometries preserve the geometric decomposition of Haken manifolds , preprint, December 29, 1995 . · Zbl 0866.20033
[7] W. S. MASSEY , A basic course in algebraic topology , New York, Springer 1991 . MR 92c:55001 | Zbl 0725.55001 · Zbl 0725.55001
[8] G. D. MOSTOW , Strong rigidity of locally symmetric spaces (Ann. of Math. Studies, No. 78, Princeton Univ. Press 1973 ). MR 52 #5874 | Zbl 0265.53039 · Zbl 0265.53039
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