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 $1$-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 $M$ is homeomorphic to the interior of a compact manifold with boundary if the manifold $M$ is not less curved than a noncompact model surface $\mathrm{M̃}$ of revolution and if the total curvature of the model surface $\mathrm{M̃}$ is finite and less than $2\pi$. 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 $G$ be a finite group, and let $k$ be a nonnegative integer. We say that $G$ has uniform spread $k$ if there exists a fixed conjugacy class $C$ in $G$ with the property that for any $k$ nontrivial elements $x_1,\dots ,x_k$ in $G$ there exists $y\in C$ such that $G=\langle x_i,y\rangle$ for all $i$. Further, the exact uniform spread of $G$, denoted by $u\left(G\right)$, is the largest $k$ such that $G$ has the uniform spread $k$ property. By a theorem of Breuer, Guralnick, and Kantor, $u\left(G\right)\ge 2$ for every finite simple group $G$. Here we consider the uniform spread of almost simple linear groups. Our main theorem states that if $G=\langle {PSL}_n\left(q\right),g\rangle$ is almost simple, then $u\left(G\right)\ge 2$ (unless $G\cong S_6$), and we determine precisely when $u\left(G\right)$ tends to infinity as $\left|G\right|$ tends to infinity.

Let $R$ be a perfect ${\mathbb{F}}_p$-algebra equipped with the trivial norm. Let $W\left(R\right)$ be the ring of $p$-typical Witt vectors over $R$ equipped with the $p$-adic norm. At the level of nonarchimedean analytic spaces (in the sense of Berkovich), we demonstrate a close analogy between $W\left(R\right)$ and the polynomial ring $R\left[T\right]$ equipped with the Gauss norm, in which the role of the structure morphism from $R$ to $R\left[T\right]$ is played by the Teichmüller map. For instance, we show that the analytic space associated to $R$ is a strong deformation retract of the space associated to $W\left(R\right)$. 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 $p$-adic representations of étale fundamental groups of nonarchimedean analytic spaces (i.e., relative $p$-adic Hodge theory).