Mathematics Colloquia and Seminars

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Assume that (M_i,p_i) is a sequence of n-manifolds satisfying Ric \geq -(n-1) and converges to (X,p) in the Gromov-Hausdorff sense. We shall show that X is semi-locally simply connected.  In particular, if M_i is simply connected and X is compact, then X is simply connected. In the case that M_i is simply connected and X is non-compact, X is not necessarily simply connected; so we further assume that there is a discrete isometric group action G_i on M_i so that the diameter of M_i/G_i is uniformly bounded, we may assume G_i equivariantly converges to an isometric group G on X. We can show that G_0, the identity component of G, is a nilpotent group and there is a maximal torus T^k in G_0. We can show \pi_1(X) is generated by the T^k orbit of p.