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Margulis lemma and Hurewicz fibration theorem on Alexandrov spaces. (English) Zbl 1502.53062

The authors generalize Margulis lemma (see, [M. Gromov, J. Differ. Geom. 13, 231–241 (1978; Zbl 0432.53020)]) to Alexandrov spaces of curvature bounded below. Their main result (Theorem 1.1) states that there are positive constants \(\varepsilon\left( n\right) \) and \(w\left( n\right) \) depending on \(n\) such that for every Alexandrov \(n\)-space \(M\) with Alexandrov’s curvature bounded below by \(-1\) and any point \(p\in M\), the subgroup \(\Gamma_{p}\left( p;\varepsilon\right) \) of the fundamental group \(\pi_{1}\left( B_{1}\left( p\right) ,p\right) \) generated by loops at \(p\) lying in \(B_{\varepsilon }\left( p\right) \) with \(\varepsilon\in\left( 0,\varepsilon\left( n\right) \right) \) is \(w\left( n\right) \)-nilpotent. The proof uses main ideas from [V. Kapovitch et al., Ann. Math. (2) 171, No. 1, 343–373 (2010; Zbl 1192.53040)].

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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