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Alberto Verjovsky

Publications and source records attributed to Alberto Verjovsky.

At least 19 recordsLinked to original sources
arXiv2026

Quaternionic Extensions of Hyperbolic Toral Automorphisms

We construct real-analytic diffeomorphisms of \(S^3\times S^3\cong \SU(2)\times\SU(2)\) that lift hyperbolic toral automorphisms by evaluating Nielsen automorphisms of the free group \(F_2\) on \(\SU(2)^2\). Every matrix in \(\mathrm{GL}(2,\mathbb Z)\) admits such a lift, and every lift preserves product Haar measure. For each maximal torus \(T\subset\SU(2)\), the product \(T\times T\) is invariant and carries the original toral dynamics; the union of these tori is exactly the commuting locus. Simultaneous conjugation turns toral periodic points into periodic conjugacy two-spheres, which rules out ambient Anosov hyperbolicity. The induced action on the \(\SU(2)\)-character variety is canonical, and on the boundary pillowcase it is the quotient of the toral automorphism by \(\xi\mapsto-\xi\). Finally, normalized forward and backward images of the coordinate three-cycles converge to stable and unstable eigen-currents supported on the commuting locus.

math.DS↗
arXiv2026

Holomorphic Linear $\C^k$-Actions, Trace Foliations, and Higher-Rank Poincar\'e Dynamics

We study the orbit decomposition on $\C^n$ generated by diagonal holomorphic $\C^k$-actions in the higher-rank setting of the classical Poincar\'e--Siegel dichotomy for linear vector fields. The coordinate stratification determines the dimensions and isotropy groups of the leaves and, under a maximal-rank condition, gives a precise description of the orbit structure on every coordinate stratum. For configurations in the Poincar\'e domain, a separating real direction provides a global conical model of the punctured orbit foliation by its traces on Euclidean spheres. We construct the corresponding radially reparametrized action on a sphere, describe its leaves as homogeneous spaces, and prove that orbit closures are constrained by coordinate supports. In particular, a limit point cannot acquire a new nonzero coordinate, although nonclosed trace leaves may also accumulate within a fixed support stratum. We show that the geometry of the weight configuration determines the complex dimensions of the leaves, whereas the arithmetic of their effective isotropy groups determines their diffeomorphism types. This gives rise to a threshold phenomenon across the coordinate stratification: the diffeomorphism type of the leaves is rigid in the low- and high-dimensional regimes, but becomes arithmetically unstable in the intermediate range $k<|I|<2k$. Finally, we show that these singular foliations admit canonical local transverse holomorphic structures in the spirit of Haefliger's transverse geometry for regular foliations. These structures determine intrinsic transverse pseudogroups, yielding a well-defined local transverse holomorphic geometry for the orbit foliation.

math.DS↗
arXiv2026

How Random Is the M\"obius Function? Smoothing, Probability, and the Riemann Hypothesis

This article is primarily expository, but it also contains several new observations and reformulations concerning the M\"obius function, probabilistic models, dynamical systems, and the Riemann hypothesis. Its starting question is classical: in what sense can the M\"obius function be said to behave randomly? We begin with Denjoy's random-walk heuristic and place it in the context of later work on random multiplicative functions, short intervals, and M\"obius pseudorandomness. We then develop two smoothing forms of the classical Mertens criterion for the Riemann hypothesis. The first uses the discrete Laplace transform \[ \Phi(t)=\sum_{n\geq1}\mu(n)e^{-nt} \] and identifies RH with the condition \[ \Phi\in L^p(0,\infty) \qquad\text{for every }1\leq p<2. \] The second uses normalized M\"obius Fourier polynomials and local moments on arcs of length comparable with \(1/N\). The paper also revisits the author's earlier criterion for RH in terms of discrete measures, as presented in Broughan's account of analytic equivalents of RH. Denjoy's heuristic is made precise in a simple independent coefficient model, and this model is carefully distinguished from the modern theory of random multiplicative functions. The resulting coefficient space also gives a measure--category contrast: the relevant \(L^p\)-property has full measure but is topologically meagre. A further point of the paper is dynamical: the multiplicative semigroup of positive integers acts naturally on the coefficient space, and the M\"obius sequence is a distinguished arithmetic point for this action. The aim throughout is explanatory, while keeping these new observations visible: to show how arithmetic, probability, Fourier analysis, Mellin transforms, and dynamics fit together, and to indicate what modern results add to the older random-walk picture.

math.PR↗
arXiv2026

Adelic Loop Groups and Perfectoid Analogies: Factorization and Holomorphic Bundles on the Adelic Projective Line

We develop a theory of adelic loop groups on the universal one-dimensional solenoid \(S^1_{\mathbb Q}=(\mathbb R\times\widehat{\mathbb Z})/\mathbb Z_{\mathrm{diag}}\), the compact abelian group whose Pontryagin dual is \(\mathbb Q\) rather than \(\mathbb Z\). We introduce the adelic projective line \(\mathbb{CP}^1_{\mathbb Q}\), its ring of Laurent--Puiseux series, and holomorphic vector bundles defined by solenoidal clutching data. We prove that its Picard group is naturally isomorphic to the additive group \(\mathbb Q\). The paper establishes a scalar Wiener--Birkhoff factorization theorem, a matrix Wiener lemma, exact factorization for ordered triangular and small-norm cocycles, a density theorem for factorable matrix loops in the Wiener algebra \(\mathfrak W_{\mathbb Q}\), and a Birkhoff--Grothendieck splitting theorem in the pro-algebraic category. These results lead to the Solenoidal Birkhoff--Grothendieck conjecture, asserting that every \(g\in \mathrm{GL}*n(\mathfrak W*{\mathbb Q})\) admits a factorization \(g=h_-^{-1}\operatorname{diag}(\chi_{q_1},\ldots,\chi_{q_n})h_+\), where \(h_\pm\in\mathrm{GL}*n(\mathfrak W^\pm*{\mathbb Q})\) and \(q_i\in\mathbb Q\). We also develop the Kahler, Grassmannian, and Morse--Bott geometry of adelic loop groups in the spirit of Pressley--Segal. Finally, we compare the theory with perfectoid geometry. The Fargues--Fontaine curve provides a non-archimedean structural counterpart of \(\mathbb{CP}^1_{\mathbb Q}\) at the level of rational slope data, Kedlaya's slope theory supplies a (p)-adic analogue of Wiener--Birkhoff factorization, and the Fargues--Fontaine classification provides a proved perfectoid model for the matrix splitting problem formulated here. This comparison yields a Harder--Narasimhan reformulation of the Solenoidal Birkhoff--Grothendieck conjecture.

math.FA↗
arXiv2026

Mostow rigidity for skew solenoidal manifolds

We prove a Mostow rigidity theorem for foliated bundles over closed hyperbolic manifolds of dimension $n \geq 3$ endowed with a completely invariant measure of full support. These include solenoidal manifolds obtained as inverse limits of directed systems of finite coverings of closed hyperbolic manifolds. This theorem then extends to skew solenoidal manifolds for which the action of the holonomy group is twisted by means of a cocycle.

math.DS↗
arXiv2026

Wild knots embedded in the Menger Sponge

In this paper, we provide explicit recursive constructions of infinitely many non-equivalent wild knots contained in the Menger sponge, in such a way that we can control their set of wild points that lies in a usual Cantor set contained in the Menger sponge. Furthermore, we show that wild knots of dynamically defined type arising from Kleinian group actions can be isotoped into the sponge. We want to emphasize that our approach is constructive and geometric.

math.GT↗
arXiv2025

A Constructive Cubical Realization of $n$-Dimensional Smooth Knots Inside the Menger $M^{n+2}_n$-continuum

We prove that every smooth $n$-dimensional knot in $\mathbb{R}^{n+2}$ can be ambiently isotoped into the Menger $n$-dimensional continuum. In contrast with classical embedding theorems for universal compacta, our construction is explicit and proceeds via cubical models, combining the cubical realization theorem of Boege--Hinojosa--Verjovsky with the affine self-similarity of the Menger continuum.

math.GT↗
arXiv2024

$N$-dimensional beaded necklaces and higher dimensional wild knots, invariant by a Schottky group

Starting with a smooth, non-trivial $n$-dimensional knot $K\subset\bS^{n+2}$, and a beaded $n$-dimensional necklace subordinated to $K$, we construct a wild knot with a Cantor set of wild points (\ie the knot is not locally flat in these points). The construction uses the conformal Schottky group acting on $\bS^{n+2}$, generated by inversions on the spheres which are the boundary of the ``beads''. We show that if $K$ is a fibered knot, then the wild knot is also fibered. We also study cyclic branched coverings along the wild knots. This work generalizes the result presented in [8].

math.GT↗
arXiv2023

Asymptotic Homology of Brownian motion on a Riemannian manifold

We prove, using the celebrated result by Spitzer about winding of planar Brownian motion, and the existence of harmonic morphisms $f:M\to{\mathbb S}^1$ representing cohomology classes in $\text{H}^1(M,\mathbb Z)$, that there is a stochastic process $H_t:{\mathcal C}(M)\to{\text{Hom}(\text{H}^1(M;\mathbb R), \mathbb R)}\simeq{\text{H}_1(M;\mathbb R)}$ ($t\in[0,\infty)$), where ${\mathcal C}(M)= \{ \alpha:[0, \infty) \to M :\alpha \,\, \text{is continuous} \}$, which has a multivariate Cauchy distribution i.e. such that for each nontrivial cohomology class $[\omega]\in{\text{H}^1(M;\mathbb R), \mathbb R)}$, represented by a closed 1-form $\omega$, in the de Rham cohomology, the process $A^\omega_t:{\mathcal C}(M)\to\mathbb R\,$ ($t\in[0,\infty)$) with $A^\omega_t(B)=H_t(B)([\omega]),\, B\in{\mathcal C}(M)$ converges in distribution, with respect to Wiener measure on ${\mathcal C}(M)$, to a Cauchy's distribution, with parameter 1. The process describes the ``homological winding" of the Brownian paths in $M$, thus it can be regarded as a generalization of Spitzer result. The last section discusses the asymptotic behavior of holonomy along Brownian paths.

math.PR↗
arXiv2023

Bianchi and Hilbert-Blumenthal quaternionic orbifolds

In a series of papers, published in Mathematische Annalen, Bianchi and Blumenthal introduced the notions of Bianchi orbifolds and Hilbert-Blumnethal surfaces as generalizations of modular curves associated to quadratic fields. In this paper, in the same spirit, and following a similar line of reasoning, we introduce the concept of Bianchi and Hilbert-Blumenthal quaternionic orbifolds as generalizations of the Lipschitz and Hurwitz quaternionic modular orbifolds defined recently by D\'iaz, Vlacci and the first author. In particular, we describe the cusp cross-sections of the Hilbert-Blumenthal quaternionic orbifolds in terms of fundamental units of real quadratic fields. These are 7-dimensional solvmanifolds which are virtual ${\mathbb T}^6$ bundles over the circle with monodromy a linear Anosov diffeomorphism of the 6-torus.

math.NT↗
arXiv2022

Low-dimensional solenoidal manifolds

In this paper we survey $n$-dimensional solenoidal manifolds for $n=1,2$ and 3, and present new results about them. Solenoidal manifolds of dimension $n$ are metric spaces locally modeled on the product of a Cantor set and an open $n$-dimensional disk. Therefore, they can be "laminated" (or "foliated") by $n$-dimensional leaves. By a theorem of A. Clark and S. Hurder, topologically homogeneous, compact solenoidal manifolds are McCord solenoids i.e. are obtained as the inverse limit of an increasing tower of finite, regular covers of a compact manifold with an infinite and residually finite fundamental group. In this case their structure is very rich since they are principal Cantor-group bundles over a compact manifold and they behave like "laminated" versions of compact manifolds, thus they share many of their properties. These objects codify the commensurability properties of manifolds.

math.DG↗
arXiv2021

Errett Bishop theorems on Complex Analytic Sets: Chow's Theorem Revisited and Foliations with all leaves Compact on K\"ahler Manifolds

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\"ahler manifolds.

math.CV↗
arXiv2021

Uniformization of compact foliated spaces by surfaces of hyperbolic type

We give a new proof of the uniformization theorem of the leaves of a lamination by surfaces of hyperbolic conformal type. We use a laminated version of the Ricci flow to prove the existence of a laminated Riemannian metric (smooth on the leaves, transversaly continuous) with leaves of constant Gaussian curvature equal to -1, which is conformally equivalent to the original metric.

math.DG↗
arXiv2021

Chow's Theorem Revisited

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.

math.AG↗
arXiv2020

Some aspects of Rotation Theory on compact abelian groups

In this paper we present a generalization of Poincar\'e's Rotation Theory of homeomorphisms of the circle to the case of one-dimensional compact abelian groups which are solenoidal groups, {\it i.e.}, groups which fiber over the circle with fiber a Cantor abelian group. We define rotation elements, \emph{\`a la} Poincar\'e and discuss the dynamical properties of translations on these solenoidal groups. We also study the semiconjugation problem when the rotation element generates a dense subgroup of the solenoidal group. Finally, we comment on the relation between Rotation Theory and entropy for these homeomorphisms, since unlike the case of the circle, for the solenoids considered here there are homeomorphisms (not homotopic to the identity) with positive entropy.

math.DS↗
arXiv2020

Some remarks on equilateral triangulations of surfaces and Belyi functions

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 $\tau$ 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.

math.CV↗
arXiv2020

Quantum (Non-commutative) Toric Geometry: Foundations

In this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering quantum toric varieties as (non-algebraic) stacks, we define their category and show that it is equivalent to a category of quantum fans. We develop a Quantum Geometric Invariant Theory (QGIT) type construction of Quantum Toric Varieties. Unlike classical toric varieties, quantum toric varieties admit moduli and we define their moduli spaces, prove that these spaces are orbifolds and, in favorable cases, up to homotopy, they admit a complex structure.

math.SG↗