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We present a proof of Chow's theorem using two results of Errett Bishop retated to volumes and limits of analytic varieties. We think this approach suggested a long time ago in the beautiful book by Gabriel Stolzenberg, is very attractive and easier for students and newcomers to understand, also the theory presented here is linked to areas of mathematics that are not usually associated with Chow's theorem. Furthermore, Bishop's results imply both Chow's and Remmert-Stein's theorems directly, meaning that this approach is more economic and just as profound as Remmert-Stein's proof. At the end of the paper there is a comparison table that explains how Bishop's theorems generalize to several complex variables classical results of one complex variable., The paper is superceeded by arXiv:2111.04846 math.CV It containes new material and has some minor corrections

In this paper we present a series of seemingly unrelated results of Complex Analysis which are in fact connected via a different approach to their proofs using the results of Errett Bishop of volumes and limits of analytic varieties. We start by proving Chow's theorem by a technique suggested long time ago in the beautiful book by Gabriel Stolzenberg. We think this approach is very attractive and easier for students and newcomers to understand; also the theory presented here is linked to areas of mathematics that are not usually associated with Chow's result. In addition, Bishop's results imply both Chow's and Remmert-Stein's theorems directly, meaning that this view is simpler and just as profound as Remmert-Stein's proof. After that, we give a comparison table that explains how Bishop's theorems generalize to several complex variables classical results of one complex variable and prove Montel's compactness theorem using the techniques presented here. Finally we give an alternative proof of a theorem of Edwards, Millet and Sullivan of foliations with compact leaves for the case of complex foliations in K��hler manifolds., arXiv admin note: substantial text overlap with arXiv:2101.09872

In this paper, following Grothendieck {\it Esquisse d'un programme}, which was motivated by Belyi's work, we study some properties of surfaces $X$ which are triangulated by (possibly ideal) isometric equilateral triangles of one of the spherical, euclidean or hyperbolic geometries. These surfaces have a natural Riemannian metric with conic singularities. In the euclidean case we analyze the closed geodesics and their lengths. Such surfaces can be given the structure of a Riemann surface which, considered as algebraic curves, are defined over $\bar{\mathbb{Q}}$ by a theorem of Belyi. They have been studied by many authors of course. Here we define the notion of connected sum of two Belyi functions and give some concrete examples. In the particular case when $X$ is a torus, the triangulation leads to an elliptic curve and we define the notion of a "peel" obtained from the triangulation (which is a metaphor of an orange peel) and relate this peel with the modulus $��$ of the elliptic curve. Many fascinating questions arise regarding the modularity of the elliptic curve and the geometric aspects of the Taniyama-Shimura-Weil theory.

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This article presents a generalization of the notion of \emph{Poincar�� rotation set} to homeomorphisms of the ad��le class group $\mathbb{A}/\mathbb{Q}$ of the rational numbers $\mathbb{Q}$, which is a connected compact abelian group which can be identified with the one-dimensional universal solenoid $\mathbf{S}$, \ie the algebraic universal covering of the circle. The definition is first introduced in general for homeomorphisms of $\mathbf{S}$ which are isotopic to a translation, and then specializing in homeomorphisms of $\mathbf{S}$ isotopic to the identity, in which case the rotation set is a closed interval contained in the base leaf (the connected component of the identity). If in the latter case the rotation interval reduces to a single element $��$ and $��$ is irrational (\ie it is a monothetic generator of $\mathbf{S}$), we show that the homeomorphism is semiconjugate to the translation $z\mapsto��{z}$, like in the classical theory of Poincar��. This theory is valid for any general compact abelian one dimensional solenoidal group $\mathbf{S}_G$, which are Pontryagin duals of dense subgroups $G$ of the rational numbers with the discrete topology. These solenoidal groups are one-dimensional laminations which are locally homeomorphic to the product of a Cantor set by an interval so they behave very much like a ``diffuse'' version of the circle. Our approach differs from others because we use Pontryagin duality of compact abelian groups to define the rotation sets. \end{abstract}, 15 pages

We present an unifying method to prove criteria for the Riemann Hypothesis involving the arithmetical functions of Euler, von Mangoldt, M��bius and Liouville.