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A convergence theorem in the geometry of Alexandrov spaces. (English) Zbl 0885.53041
Besse, Arthur L. (ed.), Actes de la table ronde de géométrie différentielle en l’honneur de Marcel Berger, Luminy, France, 12–18 juillet, 1992. Paris: Société Mathématique de France. Sémin. Congr. 1, 601-642 (1996).
An Alexandrov space is a metric space with length structure and with a notion of curvature. The corresponding definitions can be found in [Yu. D. Burago, M. Gromov and G. Perel’man, Russ. Math. Surv. 47, 1-58 (1992; Zbl 0802.53018)].
The main purpose of this paper is to extend the fibration theorem in [T. Yamaguchi, Ann. Math., II. Ser. 133, 317-357 (1991; Zbl 0737.53041)] to the case of Alexandrov spaces.
Theorem: For a given positive integer \(n\) and \(\mu_0>0\), there exist positive numbers \(\delta= \delta_n\) and \(\varepsilon= \varepsilon (\mu_0)\) satisfying the following. Let \(X\) be an \(n\)-dimensional complete Alexandrov space with curvature \(\geq -1\) and with \(\delta\)-strain radius of \(X\) greater than \(\mu_0\). Then, if the Hausdorff distance between \(X\) and a complete Alexandrov space \(M\) with curvature \(\geq 1\) is less than \(\varepsilon\), then there exists a \(\tau (\delta, \varepsilon)\)-almost Lipschitz submersion \(f:M\to X\). Here, \(\tau (\delta, \sigma)\) denotes a positive constant depending on \(n\), \(\mu_0\) and \(\delta,\varepsilon\), and satisfying \(\lim_{\delta, \varepsilon\to 0} \tau(\delta, \varepsilon)=0\).
The author conjectures that \(f\) should be a locally trivial fiber bundle. Using above Theorem, the author revisits several works of Burago and Perelman and also proves the following generalization of the Fukaya-Yamaguchi theorem for Riemannian manifolds.
Theorem: There exists a positive number \(\varepsilon_n\) such that if \(X\) is an \(n\)-dimensional compact Alexandrov space with curvature \(\geq -1\) and \(\text{diam} (X)< \varepsilon_n\), then its fundamental group contains a nilpotent subgroup of finite index.
For the entire collection see [Zbl 0859.00016].

53C20 Global Riemannian geometry, including pinching
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)