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On the structure of spaces with Ricci curvature bounded below. II. (English) Zbl 1027.53042
The reviewed paper, the sequel of the authors’ article [J. Differ. Geom. 46, 406-480 (1997; Zbl 0902.53034)], is the second in a series devoted to the study of the structure of complete connected Riemannian manifolds, whose Ricci curvature has a definite lower bound and of the Gromow-Hausdorff limits, $$Y$$, of sequences of such manifolds. First the authors prove a generalization in the collapsing case, of the volume convergence conjecture of Anderson-Cheeger and deduce some consequences. Next they introduce a family of lower dimensional Hausdorff measures associated to a renormalized limit measure, $$\nu$$, on a collapsed limit space and show that for the notion of codimension defined by this family of measures, the strongly singular set satisfies $$\text{ codim}_{\nu}\mathcal S\mathcal S\geqslant 1$$. Then the authors investigate connectedness properties of the regular set $$\mathcal R$$ of $$Y$$, which is the limit of manifolds with Ricci curvature uniformly bounded from below. Using these results they prove that the isometry group of noncollapsed limit space is a Lie group. Finally, the authors show that a limit space which contains a one 1-dimensional piece and which satisfies an additional condition, is itself 1-dimensional.
For Part III, see ibid. 54, 37-74 (2000; Zbl 1027.53043).

##### MSC:
 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53C20 Global Riemannian geometry, including pinching 49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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