# Margulis lemma and Hurewicz fibration Theorem on Alexandrov spaces

## TL;DR

This paper extends the Margulis lemma to Alexandrov spaces with curvature bounds, establishes Hurewicz fibrations for certain submersions, and improves gradient push techniques, advancing geometric and topological understanding of these spaces.

## Contribution

It generalizes the Margulis lemma with a uniform index bound and proves that regular almost Lipschitz submersions are Hurewicz fibrations, with improved gradient push bounds.

## Key findings

- Proved the generalized Margulis lemma with a uniform index bound.
- Established that Yamaguchi's regular almost Lipschitz submersions are Hurewicz fibrations.
- Improved the universal pushing time bound for gradient push.

## Abstract

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 $p\in X$ generate a subgroup of the fundamental group of unit ball $B_1(p)$ that contains a nilpotent subgroup of index $\le w(n)$, 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.