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Margulis lemma and Hurewicz fibration theorem on Alexandrov spaces

    https://doi.org/10.1142/S0219199721500486Cited by:4 (Source: Crossref)

    We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov n-space X with curvature bounded below, i.e. small loops at pX generate a subgroup of the fundamental group of the unit ball B1(p) that contains a nilpotent subgroup of index w(n), where w(n) is a constant depending only on the dimension n. The proof is based on the main ideas of V. Kapovitch, A. Petrunin and W. Tuschmann, and the following results:

    (1) We prove that any regular almost Lipschitz submersion constructed by Yamaguchi on a collapsed Alexandrov space with curvature bounded below is a Hurewicz fibration. We also prove that such fibration is uniquely determined up to a homotopy equivalence.

    (2) We give a detailed proof on the gradient push, improving the universal pushing time bound given by V. Kapovitch, A. Petrunin and W. Tuschmann, and justifying in a specific way that the gradient push between regular points can always keep away from extremal subsets.

    AMSC: 53C21, 53C23