Margulis lemma and Hurewicz fibration theorem on Alexandrov spaces
Abstract
We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov -space with curvature bounded below, i.e. small loops at generate a subgroup of the fundamental group of the unit ball that contains a nilpotent subgroup of index , where is a constant depending only on the dimension . 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.