Margulis lemma and Hurewicz fibration theorem on Alexandrov spaces

Abstract

1 We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov [Formula: see text]-space [Formula: see text] with curvature bounded below, i.e. small loops at [Formula: see text] generate a subgroup of the fundamental group of the unit ball [Formula: see text] that contains a nilpotent subgroup of index [Formula: see text], where [Formula: see text] is a constant depending only on the dimension [Formula: see text]. 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.