Margulis lemma and Hurewicz fibration theorem on Alexandrov spaces

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 p is an element of X generate a subgroup of the fundamental group of the unit hall B-1(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.

Automated summary

This article proves the generalized Margulis lemma on an Alexandrov n-space X with curvature bounded below, providing a bound on the index of nilpotent subgroups in the fundamental group. It also discusses regular almost Lipschitz submersions constructed by Yamaguchi and gives a detailed proof on gradient push.