arXiv research

The paper classifies singularities of line congruences in 4D space.

The paper classifies singularities of plane congruences and affine distance functions.

An explicit construction of surfaces with flat normal bundle in the Euclidean space (unit hypersphere) in terms of solutions of certain linear system is proposed. In the case of 3-space our formulae can be viewed as the direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geom…

The study examines singularities and geometric properties of surfaces derived from frontals with specific singular points.

Extends Kummer's theory to singular surfaces for line congruences.

We study the geodesic flow on the normal line congruence of a minimal surface in ${\Bbb{R}}^3$ induced by the neutral Kähler metric on the space of oriented lines. The metric is lorentz with isolated degenerate points and the flow is shown to be completely integrable. In addition, we give a new holomorphic description …

We describe natural Kähler or para-Kähler structures of the spaces of geodesics of pseudo-Riemannian space forms and relate the local geometry of hypersurfaces of space forms to that of their normal congruences, or Gauss maps, which are Lagrangian submanifolds. The space of geodesics L(S^{n+1}_{p,1}) of a pseudo-Rieman…

The study characterizes subgroups of braid groups and their finite quotients.

Let A denote the algebraic closure of the rationals Q in the complex numbers C. Suppose G is a torsion-free group which contains a congruence subgroup as a normal subgroup of finite index and denote by U(G) the C-algebra of closed densely defined unbounded operators affiliated to the group von Neumann algebra. We prove…

We discuss a number of open problems about mapping class groups of surfaces. In particular, we discuss problems related to linearity, congruence subgroups, cohomology, pseudo-Anosov stretch factors, Torelli subgroups, and normal subgroups.

The study identifies unique fluid flow patterns.

We prove that a deformation of a hypersurface in a $(n+1)$-dimensional real space form ${\mathbb S}^{n+1}_{p,1}$ induce a Hamiltonian variation of the normal congruence in the space ${\mathbb L}({\mathbb S}^{n+1}_{p,1})$ of oriented geodesics. As an application, we show that every Hamiltonian minimal sumbanifold in ${\…

We consider the isometric deformation problem for oriented non simply connected immersed minimal surfaces $f:M \to S^{4}$. We prove that the space of all isometric minimal immersions of $M$ into $S^{4}$ with the same normal curvature function is, within congruences, either finite or a circle. Furthermore, we show that …

Given an oriented Riemannian surface $(Σ, g)$, its tangent bundle $TΣ$ enjoys a natural pseudo-Kähler structure, that is the combination of a complex structure $\J$, a pseudo-metric $\G$ with neutral signature and a symplectic structure $\Om$. We give a local classification of those surfaces of $TΣ$ which are both Lagr…

Study on braid group quotients by congruence subgroups.

The paper describes the geometric properties of line congruences' singularities.

Study on braid groups' congruence subgroups and their crystallographic quotients.

Let f be an integer greater than one. We study three progressively finer equivalence relations on closed 3-manifolds generated by Dehn surgery with denominator f: weak f-congruence, f-congruence, and strong f-congruence. If f is odd, weak f-congruence preserves the ring structure on cohomology with Z_f-coefficients. We…

This paper explores geometric insights into discrete R-congruences and their envelopes.

We enumerate all the principal congruence link complements in $S^3$, there by answering a question of W. Thurston. Related articles: "Technical Report: All Principal Congruence Link Groups" (arXiv:1902.04722), "All Known Principal Congruence Links" (arXiv:1902.04426).

Paper explores relations between braid groups and their quotients.

We study the moduli space of congruence classes of isometric surfaces with the same mean curvature in 4-dimensional space forms. Having the same mean curvature means that there exists a parallel vector bundle isometry between the normal bundles that preserves the mean curvature vector fields. We prove that if both Gaus…

New BDEs reveal singular surfaces from line congruences.

The study explores congruence subgroups of braid groups and their quotients.

Study of line congruences for Appell's rank-4 hypergeometric functions.

We give a sufficient condition for isometric actions to have the congruency of orbits, that is, all orbits are isometrically congruent to each other. As applications, we give simple and unified proofs for some known congruence results, and also provide new examples of isometric actions on symmetric spaces of noncompact…

Characterizes W-congruences to study their stable umbilical points.

We continue the investigation of the correspondence between systems of conservation laws and congruences of lines in projective space. Relationship between "additional" conservation laws and hypersurfaces conjugate to a congruence is established. This construction allows us to introduce, in a purely geometric way, the …

Proves congruence subgroup property for mapping class groups of hyperbolic surfaces.

We study "how far away" a finite index subgroup G of SL(2,Z) is from being a congruence group. For this we define its deficiency of being a congruence group. We show that the index of the image of G in SL(2,Z/nZ) is biggest, if n is the general Wohlfahrt level. We furthermore show that the Veech groups of origamis (or …

Let S be a smooth affine algebraic curve, and let S' be the Riemann surface obtained by removing a point from S. We provide evidence for the congruence subgroup property of the mapping class group Mod(S') by showing that its congruence kernel lies in the centralizer of every braid in Mod(S'). As a corollary, we obtain …

The paper explores affine geometry of line congruences using singularity theory.

It is known that the level $2$ principal congruence subgroup of $GL(n;\mathbb{Z})$ has a finite generating set. In this paper, we give a finite presentation of the level $2$ principal congruence subgroup of $GL(n;\mathbb{Z})$.

We study Veech groups of covering surfaces of primitive translation surfaces. Therefore we define congruence subgroups in Veech groups of primitive translation surfaces using their action on the homology with entries in $\mathbb{Z}/a\mathbb{Z}$. We introduce a congruence level definition and a property of a primitive t…

Study virtual braid groups, proving a key subgroup result.

Proves Congruence Subgroup Property for two types of groups.

The paper studies congruence subgroups and crystallographic quotients of small Coxeter groups.

We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class gro…

There is a natural duality between line congruences in $\mathbb{R}^3$ and surfaces in $\mathbb{R}^4$ that sends principal lines into asymptotic lines. The same correspondence takes the discriminant curve of a line congruence into the parabolic curve of the dual surface. Moreover, it takes the ridge curves to the flat r…

One of the well-known challenges in computer vision tasks is the visual diversity of images, which could result in an agreement or disagreement between the learned knowledge and the visual content exhibited by the current observation. In this work, we first define such an agreement in a concepts learning process as con…

We give some new congruences for singular real algebraic curves which generalize Fiedler's congruence for nonsingular curves.

Theorem proves congruence for compact submanifolds in a sphere.

This is a technical report accompanying the paper "All Principal Congruence Link Groups" (arXiv:1802.01275) classifying all principal congruence link complements in S^3 by the same authors. It provides a complete overview of all cases (d,I) that had to be considered, as well as describes the necessary computations and …

Ng and Schauenburg proved that the kernel of a $(2+1)$-dimensional topological quantum field theory representation of $\mathrm{SL}(2, \mathbb{Z})$ is a congruence subgroup. Motivated by their result, we explore when the kernel of an irreducible representation of the braid group $B_3$ with finite image enjoys a congruen…

Unified framework for studying Torelli group and congruence subgroup maps.

Let $H<\mathrm{PSL}_2(\mathbb{Z})$ be a finite index normal subgroup which is contained in a principal congruence subgroup, and let $Φ(H)\neq H$ denote a term of the lower central series or the derived series of $H$. In this paper, we prove that the commensurator of $Φ(H)$ in $\mathrm{PSL}_2(\mathbb{R})$ is discrete. W…

We describe a criterion for a real or complex hyperbolic lattice to admit a RFRS tower that consists entirely of congruence subgroups. We use this to show that certain Bianchi groups $\mathrm{PSL}(\mathcal{O}_d)$ are virtually fibered on congruence subgroups, and also exhibit the first examples of RFRS Kähler groups th…

The asymptotic lattices and their transformations are studied within the line geometry approach. It is shown that the discrete asymptotic nets are represented by isotropic congruences in the Plucker quadric. On the basis of the Lelieuvre-type representation of asymptotic lattices and of the discrete analog of the Mouta…