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 ∈ X generate a subgroup of the fundamental group of the unit ball B1(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.

中文翻译：

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亚历山德罗夫空间上的 Margulis 引理和 Hurewicz 纤维化定理

我们证明广义 Margulis 引理具有统一索引约束在 Alexandrovn-空间X曲率有界以下，即小环在p ∈ X生成单位球的基本群的子群乙1(p)包含指数的幂零子群 ≤ w(n)， 在哪里w(n)是一个常数，仅取决于维度n. 证明基于 V. Kapovitch、A. Petrunin 和 W. Tuschmann 的主要思想，并得到以下结果： (1) 我们证明了 Yamaguchi 在下面曲率有界的塌缩 Alexandrov 空间上构造的任何规则几乎 Lipschitz 淹没是Hurewicz 纤维化。我们还证明了这种纤维化是由同伦等价唯一确定的。(2) 我们给出了梯度推动的详细证明，改进了 V. Kapovitch、A. Petrunin 和 W. Tuschmann 给出的普遍推动时间界限，并以特定的方式证明了规则点之间的梯度推动可以始终保持距离来自极值子集。