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We call a Gromov–Hausdorff limit of complete Riemannian manifolds with a lower bound of Ricci curvature a Ricci limit space. Furthermore, we prove that any Ricci limit space has integral Hausdorff dimension, provided that its Hausdorff dimension is not greater than 2. We also classify -dimensional Ricci limit spaces.
We construct distinctive surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we prove that a complete noncompact Riemannian manifold is homeomorphic to the interior of a compact manifold with boundary if the manifold is not less curved than a noncompact model surface of revolution and if the total curvature of the model surface is finite and less than . By the first result mentioned above, the second result covers a much wider class of manifolds than that of complete noncompact Riemannian manifolds whose sectional curvatures are bounded from below by a constant.
Let be a finite group, and let be a nonnegative integer. We say that has uniform spread if there exists a fixed conjugacy class in with the property that for any nontrivial elements in there exists such that for all . Further, the exact uniform spread of , denoted by , is the largest such that has the uniform spread property. By a theorem of Breuer, Guralnick, and Kantor, for every finite simple group . Here we consider the uniform spread of almost simple linear groups. Our main theorem states that if is almost simple, then (unless ), and we determine precisely when tends to infinity as tends to infinity.
Let be a perfect -algebra equipped with the trivial norm. Let be the ring of -typical Witt vectors over equipped with the -adic norm. At the level of nonarchimedean analytic spaces (in the sense of Berkovich), we demonstrate a close analogy between and the polynomial ring equipped with the Gauss norm, in which the role of the structure morphism from to is played by the Teichmüller map. For instance, we show that the analytic space associated to is a strong deformation retract of the space associated to . We also show that each fiber forms a tree under the relation of pointwise comparison, and we classify the points of fibers in the manner of Berkovich’s classification of points of a nonarchimedean disk. Some results pertain to the study of -adic representations of étale fundamental groups of nonarchimedean analytic spaces (i.e., relative -adic Hodge theory).