  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.
