Let $\Sigma$ be a set of prime numbers which is either of cardinality one or equal to the set of all prime numbers. In this paper, we prove that various objects that arise from the geometry of the configuration space of a hyperbolic curve over an algebraically closed field of characteristic zero may be reconstructed group-theoretically from the pro-$\Sigma$ fundamental group of the configuration space. Let $X$ be a hyperbolic curve of type $(g,r)$ over a field $k$ of characteristic zero. Thus, $X$ is obtained by removing from a proper smooth curve of genus $g$ over $k$ a closed subscheme [i.e., the "divisor of cusps"] of $X$ whose structure morphism to Spec$(k)$ is finite étale of degree $r$; $2g-2+r>0$. Write $X_n$ for the $n$-th configuration space associated to $X$, i.e., the complement of the various diagonal divisors in the fiber product over $k$ of $n$ copies of $X$. Then, when $k$ is algebraically closed, we show that the triple $(n,g,r)$ and the generalized fiber subgroups—i.e., the subgroups that arise from the various natural morphisms $X_n \to X_m$ $[m$ < $n]$, which we refer to as generalized projection morphisms—of the pro-$\Sigma$ fundamental group $\Pi_n$ of $X_n$ may be reconstructed group-theoretically from $\Pi_n$ whenever $n \geq 2$. This result generalizes results obtained previously by the first and third authors and A. Tamagawa to the case of arbitrary hyperbolic curves [i.e., without restrictions on $(g,r)$]. As an application, in the case where $(g,r)= (0,3)$ and $n \geq 2$, we conclude that there exists a direct product decomposition

Out$(\Pi_n) = {\mathrm{GT}^{\Sigma}} \times \mathfrak{S}_{n + 3}$

—where we write "Out$(-)$" for the group of outer automorphisms [i.e., without any auxiliary restrictions!] of the profinite group in parentheses and ${\mathrm{GT}^{\Sigma}}$ (respectively, $\mathfrak{S}_{n + 3}$) for the pro-$\Sigma$ Grothendieck-Teichmüller group (respectively, symmetric group on $n + 3$ letters). This direct product decomposition may be applied to obtain a simplified purely group-theoretic equivalent definition—i.e., as the centralizer in ${\rm Out}(\Pi_n)$ of the union of the centers of the open subgroups of ${\rm Out}(\Pi_n)$—of ${\mathrm{GT}^{\Sigma}}$. One of the key notions underlying the theory of the present paper is the notion of a pro-$\Sigma$ log-full subgroup—which may be regarded as a sort of higher-dimensional analogue of the notion of a pro-$\Sigma$ cuspidal inertia subgroup of a surface group—of $\Pi_n$. In the final section of the present paper, we show that, when $X$ and $k$ satisfy certain conditions concerning "weights", the pro-$l$ log-full subgroups may be reconstructed group-theoretically from the natural outer action of the absolute Galois group of $k$ on the geometric pro-$l$ fundamental group of $X_n$.

In this paper, we prove extensions of Bonnet-Myers-type theorems obtained by Calabi and Cheeger-Gromov-Taylor via Bakry-Emery Ricci curvature.

We obtain a result about the relation between the Thurston boundary and the relatively hyperbolic boundary of Teichmüller space. Precisely, we prove that the identity map on Teichmüller space extends to a continuous surjective map from the subset of the Thurston boundary consisting of minimal measured foliations to the relatively hyperbolic boundary. As an application, to relate the Thurston compactification and the Teichmüller compactification of Teichmüller space, we construct a new compactification of Teichmüller space which is weaker than the Thurston compactification and the Teichmüller compactification.

Let $S$ be the set of all transcendental entire functions of the form $P(z) \exp (Q(z))$, where $P$ and $Q$ are polynomials. In this paper, by using the theory of polynomial-like mappings, we construct various kinds of functions in $S$ with irrationally indifferent fixed points as follows:

(1) We construct functions in $S$ with bounded type Siegel disks centered at points other than the origin bounded by quasicircles containing critical points. This is an extension of Zakeri's result in [24] for $f \in S$.

(2) We construct functions in $S$ with Cremer points whose multipliers satisfy some Cremer's condition in [6] only for rational functions. Our method shows that this condition can be applicable even in some transcendental cases.

(3) For any integer $d \geq 2$ and some $c \in {\mathbf C} \setminus \{0\}$, we show that the function of the form $e^{2\pi i \theta}z(1 + cz)^{d-1}e^z\,(\theta \in {\mathbf R} \setminus {\mathbf Q})$ has a Siegel point at the origin if and only if $\theta$ is a Brjuno number. This is an extension of Geyer's result in [11].

(4) For the function of the form $(e^{2\pi i\theta}z+\alpha z^2)e^z \,(\theta \in {\mathbf R} \setminus {\mathbf Q}, \alpha \in {\mathbf C} \setminus \{0\})$, we show that if $\alpha$ and $\theta$ satisfy some condition, then the Siegel disk centered at the origin is bounded by a Jordan curve containing a critical point, which is not a quasicircle. Moreover, we can choose $\alpha$ and $\theta$ so that the Lebesgue measure of the Julia set is positive and can also choose them so that it is zero. This is an extension of Keen and Zhang's result in [13].

In our previous work, we have generalized the notion of dually flat or Hessian manifold to quasi-Hessian manifold; it admits the Hessian metric to be degenerate but possesses a particular symmetric cubic tensor (generalized Amari-Centsov tensor). Indeed, it naturally appears as a singular model in information geometry and related fields. A quasi-Hessian manifold is locally accompanied with a possibly multi-valued potential and its dual, whose graphs are called the $e$-wavefront and the $m$-wavefront respectively, together with coherent tangent bundles endowed with flat connections. In the present paper, using those connections and the metric, we give coordinate-free criteria for detecting local diffeomorphic types of $e/m$-wavefronts, and then derive the local normal forms of those (dual) potential functions for the $e/m$-wavefronts in affine flat coordinates by means of Malgrange's division theorem. This is motivated by an early work of Ekeland on non-convex optimization and Saji-Umehara-Yamada's work on Riemannian geometry of wavefronts. Finally, we reveal a relation of our geometric criteria with information geometric quantities of statistical manifolds.

We work over an algebraically closed field of characteristic zero. A nonbirational center of a projective variety is a point from which the variety is projected nonbirationally onto its image, whose locus plays an important role in a study of the double-point divisors and the defining equations of the variety. The purpose of this paper is to show that a scroll over a curve with some conditions has no nonbirational centers. Consequently such a nondegenerate scroll is cutting out by hypersurfaces of degree $d-e + 1$ for its degree $d$ and codimension $e$ in the projective space. On the other hand, examples of scrolls over curves with nonbirational centers are constructed.