Semi-graphs of anabelioids of PSC-type and their PSC-fundamental groups (i.e., a combinatorial Galois-category-theoretic abstraction of pointed stable curves over algebraically closed fields of characteristic zero and their fundamental groups) are central objects in the study of combinatorial anabelian geometry. In the present series of papers, which consists of two successive works, we investigate combinatorial anabelian geometry of (not necessarily bijective) continuous homomorphisms between PSC-fundamental groups. This contrasts with previous researches, which focused only on continuous isomorphisms. More specifically, our main results of the present series of papers roughly state that, if a continuous homomorphism between PSC-fundamental groups is compatible with certain outer representations, then it satisfies a certain "group-theoretic compatibility property", i.e., the property that each of the images via the continuous homomorphism of certain VCN-subgroups of the domain are included in certain VCN-subgroups of the codomain. Such results may be considered as Hom-versions of the combinatorial version of the Grothendieck conjecture established in some previous works. As in the case of previous works (i.e., the Isom-versions), the proof requires different techniques depending on the types of outer representations under consideration. In the present paper, we will treat the case where the outer representations under consideration are assumed to be "$l$-graphically full", i.e., to satisfy a certain condition concerning "weights" considered with respect to the "$l$-adic cyclotomic character", where $l$ is a certain prime number. In addition, to prepare for this purpose, we include detailed expositions on "reduction techniques", namely, techniques of reduction to the maximal pro-$\Sigma$ quotients and to the abelianizations of (various open subgroups of) the PSC-fundamental groups under consideration, where $\Sigma$ is a certain set of prime numbers. Though the discussions of these "reduction techniques" are all essentially well-known to experts, we present the results in a highly unified/generalized fashion.
A hyperbolic polyhedron is called $\pi/3$-equiangular if all its dihedral angles are equal to $\pi/3$. We find a sequence $\{P_n\}$ of $\pi/3$-equiangular polyhedra different from the sequence Atkinson found in [4]. Atkinson [4] showed that ideal regular tetrahedron $P_1$ has the smallest volume among all $\pi/3$-equiangular hyperbolic polyhedra. In this paper, we show that ideal regular cube $P_2$ has the second smallest volume and pentagonal prism has the third smallest volume among $\pi/3$-equiangular polyhedra. Moreover, we have shown that the reflection groups obtained from $\{P_n\}$ are arithmetic and that of pentagonal prism is non-arithmetic.
In the present paper, we continue our study, which was initiated in the previous paper of the present series of papers, of combinatorial anabelian geometry of (not necessarily bijective) continuous homomorphisms between PSC-fundamental groups of semi-graphs of anabelioids of PSC-type. In particular, we continue to study certain Hom-versions of the combinatorial versions of the Grothendieck conjecture established in some previous works, i.e., to study certain sufficient conditions for certain group-theoretic compatibility properties described in terms of outer representations. The outer representations we mainly concern in the present paper are of PIPSC-type and of NN-type, both of which are of substantial importance in the study of algebro-geometric anabelian geometry of configuration spaces of hyperbolic curves. We also include, as a preparation for one of the main results, a presentation of a "reduction technique", namely, a technique of reduction to the "compactified quotients" of (various open subgroups of) the PSC-fundamental groups under consideration, in a similar vein to the previous paper where we included other two "reduction techniques".
In the paper, we investigate the uniqueness of an entire function of relatively slow growth as it shares a small entire function with a linear differential polynomial. The investigation is inspired by a conjecture of R. Brück.
In this paper, we give a unicity theorem of two holomorphic maps on two open Riemann surfaces with conformal metrics, which share a family of hypersurfaces $\{Q_{j}\}_{j=1}^{q}$ in $\mathbf{P}^{n}(\mathbf{C})$ located in $N$-subgeneral position. Furthermore, motivated by the value distribution properties of the Gauss maps for $K$-quasiregular harmonic surfaces in [5], we give a unicity result for $K$-quasiregular harmonic surfaces in $\mathbf{R}^{3}$.
In previous papers [19, 17], we computed the twisted Alexander polynomials and adjoint twisted Alexander polynomials for nonabelian SL2(C)-representations of genus one two-bridge knots. In this paper, we investigate their properties. When the SL2(C)-representation of a genus one two-bridge knot is parabolic, we show that (1) the coefficient of the highest degree term of the adjoint twisted Alexander polynomial is a rational multiple, independent of the representation, of the square of the twisted Alexander polynomial at t = 1, and (2) the adjoint twisted Alexander polynomial is a monic polynomial if and only if the knot is fibered.
In the present paper, we study continuous open homomorphisms between the Galois groups of solvably closed Galois field extensions of number fields. In particular, we discuss Uchida's conjecture that asserts that an arbitrary continuous open homomorphism between the Galois groups of solvably closed Galois field extensions of number fields arises from a homomorphism between the given Galois field extensions. In the present paper, we prove that this conjecture is equivalent to the assertion that if the Galois group of a Galois field extension of a number field is isomorphic to an open subgroup of the maximal prosolvable quotient of the absolute Galois group of the field of rational numbers, then, for all prime numbers l and all but finitely many prime numbers p, the given Galois extension field contains l roots of the polynomial t(l)-p. Moreover, we prove that this conjecture is also equivalent to the assertion that if the Galois group of a Galois field extension of an absolutely Galois number field is isomorphic to an open subgroup of the maximal prosolvable quotient of the absolute Galois group of the field of rational numbers, then the given Galois extension field is absolutely Galois.
In this paper, we focus on a marginally trapped submanifold $f:\Sigma^n \to M^{n+2}_1$ in a Lorentzian manifold $M^{n+2}_1$. We show that $f$ lies in a certain null hypersurface $\mathcal{N}_f^{n+1}$ in $M^{n+2}_1$ and $f$ has a locally volume-maximizing property in $\mathcal{N}_f^{n+1}$ if $M^{n+2}_1$ satisfies the null energy condition.
We study moderate toric resolutions introduced by Chávez-Martínez, Duarte and Yasuda, which appears in the relation between F-blowups and essential divisors. In particular, we address the problems, when it exists, and if it is the case, what properties it has in conjunction with the birational geometry and Hilbert basis resolutions, mainly in dimension three.
Since the Teichmu & uml;ller space of a surface R is a deformation space of complex structures defined on R, its Bers boundary describes the degeneration of complex structures in a certain sense. In this paper, by constructing a concrete example, we prove that if R is a Riemann surface of infinite type, then there exists a Riemann surface with a marking on the Bers boundary that is homeomorphic to the surface R. We also show that such points form an infinite-dimensional complex manifold on the Bers boundary.
We provide a classification of complete improper affine spheres with singularities (say improper a parts per thousand ne fronts) in unimodular affine three-space R3 whose total curvature is greater than or equal to-6p, and a partial classification in the case of total curvature-8p. For the case of total curvature-8p, we give a complete classification for genus 0 case and show the existence of an example and a one parameter family with genus 1. We also study the asymptotic behavior of embedded ends of complete improper affine fronts. Moreover, we give new examples for this class of surfaces, including one which satisfies the equality condition of an Osserman-type inequality and is of positive genus.
We show that trivial extensions of gentle tree algebras are exactly Brauer tree algebras without exceptional vertex. We also give a characterization for the algebras whose trivial extensions are Brauer line/star/cycle algebras. As a consequence, the number of support r-tilting modules over the trivial extension T(A) of a gentle tree algebra A depends only on the number of simple A-modules.
Fargues and Scholze proved the geometric Satake equivalence over the Fargues-Fontaine curve. On the other hand, Zhu proved the geometric Satake equivalence using a Witt vector affine Grassmannian. In this paper, we explain the relation between the two versions of the geometric Satake equivalence via nearby cycle.
We introduce a new kind of welding of compact bordered Riemann surfaces, called a self-welding. We develop its fundamental theory, and apply the results to investigate, in the frame of Teichmuller theory, the set M(R-0) of marked closed Riemann surfaces of positive genus into which a given marked finite open Riemann surface R-0 of the same genus can be conformally embedded. We characterize its boundary M(R-0) in terms of self-welding closings of R-0.
We are concerned with the stabilizer poset of linear actions of finite groups. This is originally motivated through our attempt to describe the explicit geometry of the universal families over moduli spaces of Riemann surfaces. Here these universal families are locally approximated by linear quotient families associated with the linear actions of the automorphism groups of Riemann surfaces on the vector spaces of holomorphic quadratic differentials. To describe such families, the stabilizers for these linear actions play an important role. For instance, in these families, the fibers over stabilizer-constant loci are identical (the quotient fiber theorem). We in fact study the stabilizer posets, because they correspond to the posets of stabilizer-constant loci under the geometric Galois correspondence. We provide an algorithm to determine these stabilizer posets-in fact it works for the stabilizer posets for any linear action of any finite group. This algorithm is based on linear algebra combined with maximal conjugacy classes of stabilizers and is quite powerful in practical computation.
In this paper, we prove that hypersurface M-r(n) with proper mean curvature vector field (i.e. Delta(H) over right arrow is proportional to (H) over right arrow) and at most two distinct principal curvatures in a non-flat pseudo-Riemannian space form N-s(n+1) (c) is minimal or locally isoparametric, and compute the mean curvature for the isoparametric ones. As an application, we give full classification results of such non-minimal Lorentzian hypersurfaces of non-flat Lorentz space forms.
Let R and S be rings and (R)C(S )a semidualizing bimodule, and let T be a subcategory of the Auslander class A(C)(S) and H = {C circle times(S) T | T is an element of T}. Then for any left R-module M, the T-projective dimension of HomR(C, M) is at most the H-projective dimension of M, and they are identical when M is in the Bass class BC(R). If C-R(S) is faithful and T is resolving, then in a short exact sequence of left R-modules, the H-projective dimensions of any two terms can determine an upper bound of that of the third term. Furthermore, we apply these results to the cases of T being the subcategories of (weak) flat modules, projective modules and A(C)(S) respectively. Some known results are obtained as corollaries.
Let R be a Gorenstein artin algebra, and let U be a fixed left R-module and k >= 0. When U is Gorenstein injective, if the U-codominant dimension of any injective left R-module is at most k + 1, then the U-dominant dimension of any projective left R-module is at most k + 1. Dually, when U is Gorenstein projective, if the U-dominant dimension of any projective left R-module is at most k + 1, then the U-codominant dimension of any injective left R-module is at most k + 1.
Let k be an even integer and f is an element of S-K Gamma(0)(N) be a newform of weight k. We prove that for each integer N >= 16 and for k >= k(N,is an element of) (depending on N), all of nonzero zeros of the odd period polynomial associated to the newform f are on the circle |Z| = 1 root N . For each integer 3 <= N <= 15, we also investigate the location of the zeros of the odd period polynomial associated to the newform f.
