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Margulis lemma and Hurewicz fibration theorem on Alexandrov spaces

Shicheng Xu

School of Mathematical Sciences, Capital Normal University, Beijing, P. R. China

E-mail Address: shichengxu@gmail.com

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and

Xuchao Yao

Institute of Mathematics, Chinese Academy of Sciences, Beijing, P. R. China

E-mail Address: qiujuli@vip.sina.com

Corresponding author.

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

Abstract

We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov $n$ $n$ -space $X$ $X$ with curvature bounded below, i.e. small loops at $p \in X$ $p \in X$ generate a subgroup of the fundamental group of the unit ball $B_{1} (p)$ $B_{1} (p)$ that contains a nilpotent subgroup of index $\leq w (n)$ $\leq w (n)$ , where $w (n)$ $w (n)$ is a constant depending only on the dimension $n$ $n$ . 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.

Keywords:

AMSC: 53C21, 53C23

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