Study on Killing magnetic curves in Heisenberg group geometry.
problem Understanding Killing magnetic curves in Heisenberg group.
method Presentation of Heisenberg group geometry and geodesics, study of Killing magnetic curves with explicit formulas.
result Explicit formulas for Killing magnetic curves in Heisenberg group.
ODE trajectories become abnormal curves in Carnot groups.
problem Understanding abnormal curves in Carnot groups.
method Explicit construction of covectors for abnormal curves.
result Polynomial ODE trajectories lift to abnormal curves in Carnot groups.
Characterizes Coxeter groups with specific boundary shapes.
problem Identifying Coxeter groups with Sierpiński or Menger curve boundaries.
method Combining results from the literature on Gromov boundaries and Coxeter groups.
result Complete characterizations of hyperbolic Coxeter groups with Sierpiński or Menger curve boundaries.
In Carnot groups, directional pliability allows curve extensions and approximations.
problem Existence of curve extensions and approximations in Carnot groups.
method Directional pliability in subsets of directions guarantees Whitney-type extensions and Lusin approximations.
result Every horizontal curve in the Engel group intersects a C1 curve in a set of positive measure. Automorphisms of fine curve graph match surface homeomorphisms.
problem Understanding automorphisms of curve graphs for surfaces.
method Building on previous work, proving isomorphism to surface homeomorphisms.
result The group of automorphisms of the fine curve graph is isomorphic to the extended mapping class group of the surface.
In this paper we study the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group. We prove that all of the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group are helices. Moreover, we obtain explicit parametric equations for non-geodesic non-null biharmon…
Study infinite superelliptic curves and their Veech groups, providing geometric and algebraic insights.
problem Characterize Veech groups of infinite superelliptic curves.
method Analyzing geometric properties, differential equations, and group theory.
result Veech groups of infinite superelliptic curves are all matrices permuting branched points.
Study natural and conjugate mates of Frenet curves in Lie groups.
problem Characterize Frenet curves and their mates in Lie groups.
method Introduced natural and conjugate mates, derived relationships, analyzed specific curves.
result Obtained results for various Frenet curves in Lie groups.
In this paper we investigate Uludag's method for constructing new curves whose fundamental groups are central extensions of the fundamental group of the original curve by finite cyclic groups. In the first part, we give some generalizations to his method in order to get new families of curves with controlled fundamenta…
Automorphism group of nonorientable surface curve graph matches surface homeomorphisms.
problem Identifying automorphisms of nonorientable surface curve graphs.
method Using Bowden, Hensel, and Webb's fine curve graph and Long, Margalit, Pham, Verberne, and Yao's proof as a foundation.
result Automorphism group of nonorientable surface curve graph is isomorphic to the surface's homeomorphism group.
Analytic curves linked to algebraic ones via Schottky groups.
problem Moving between analytic and algebraic representations of Riemann surfaces.
method Identifying Riemann surfaces with Schottky groups and constructing families of non-hyperelliptic surfaces.
result Construction of families of non-hyperelliptic surfaces with specific properties.
A simplicial complex is called negatively curved if all its simplices are isometric to simplices in hyperbolic space, and it satisfies Gromov's Link Condition. We prove that, subject to certain conditions, a compact graph of spaces whose vertex spaces are negatively curved 2-complexes, and whose edge spaces are points …
A Carnot group G admits Lusin approximation for horizontal curves if for any absolutely continuous horizontal curve γ in G and ε>0, there is a C1 horizontal curve Γ such that Γ=γ and Γ′=γ′ outside a set of measure at most ε. We verify this property for free Carno…
New group-theoretic Johnson classes applied to curves with torsion Ceresa classes.
problem Analyzing torsion in Ceresa classes of curves.
method Group-theoretic analogues of Johnson/Morita cocycles applied to pro-l etale fundamental groups of curves.
result Example of a non-hyperelliptic curve with torsion Ceresa class.
The boundary of certain hyperbolic groups is like a Menger curve.
problem Characterizing boundaries of hyperbolic Coxeter groups.
method Analyzing the nerve of hyperbolic right-angled Coxeter groups.
result Many triangulations and disks have boundaries homeomorphic to the Menger curve.
Curves in Carnot groups avoid compact sets, growing at least t1/s.
problem Existence of periodic normal geodesics in subFinsler Carnot groups.
method Analysis of curves satisfying Pontryagin Maximum Principle.
result Normal curves in subFinsler Carnot groups leave every compact set.
We develop the structure theory of full isometry groups of locally compact non-positively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric spaces and Bruhat--Tits buildings. Applications to discrete groups and further dev…
In this paper, we give the defination of harmonic curvature function some special curves such as helix, slant curves, Mannheim curves and Bertrand curves. Then, we recall the characterizations of helices [8], slant curves (see [19]) and Mannheim curves (see [12]) in three dimensional Lie groups using their harmonic cur…
The paper characterizes simple closed curves on surfaces using profinite rigidity.
problem Characterizing simple closed curves on surfaces using profinite rigidity.
method Proving that elements with the same images under all finite groups are simple closed curves.
result The set of simple closed curves is closed in the profinite topology of the surface group.
We study the Picard groups of moduli spaces of smooth complex projective curves that have a group of automorphisms with a prescribed topological action. One of our main tools is the theory of symmetric mapping class groups. In the first part of the paper, we show that, under mild restrictions, the moduli spaces of smoo…
We survey various Alexander-type invariants of plane curve complements, with an emphasis on obstructions on the type of groups that can arise as fundamental groups of complements to complex plane curves. Also included are some new computations of higher-order degrees of curves, which are invariants defined in a previou…
Stable cylinders found in hyperbolic groups and curve graphs.
problem Torsionfree hyperbolic groups and curve graphs of surfaces have globally stable cylinders.
method Generalised Sageev's construction to improve fine properties of hyperbolic spaces.
result Proved curve graphs of surfaces admit equivariant quasi-isometric embeddings in finite products of quasitrees.
By the work of Harer, the reduced homology of the complex of curves is a fundamental cohomological object associated to all torsion free finite index subgroups of the mapping class group. We call this homology group the Steinberg module of the mapping class group. It was previously known that the curve complex has the …
The paper studies spaces of non-compact real algebraic curves and their uniformisation.
problem Understanding the spaces of non-compact real algebraic curves and their uniformisation.
method Construction of spaces of non-compact real algebraic curves and description of their connected components using Fuchsian groups.
result Any connected component of the spaces of non-compact real algebraic curves is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group.
In this paper, we show that a nontrivial compact graph manifold is nonpositively curved if and only if its fundamental group virtually embeds into a right-angled Artin group. As a consequence, nonpositively curved graph manifolds have linear fundamental groups.
Classifies special homogeneous curves with polynomial equations.
problem Identifying and classifying special homogeneous curves.
method Analyzing homogeneous polynomials and their level sets with group actions.
result All special homogeneous curves are classified.
The computation of the fundamental group of the complement of an algebraic plane curve has been theoretically solved since Zariski-van Kampen, but actual computations are usually cumbersome. In this work, we describe the notion of Wirtinger presentation of such a group relying on the real picture of the curve and with …
The paper discusses algorithms for reconstructing curves with given Euclidean or affine curvatures.
problem Reconstructing planar curves with specified Euclidean or affine curvatures.
method The paper presents algorithms for curve reconstruction under the special Euclidean and equi-affine groups.
result The reconstructed curves are close to the original curves in terms of the specified curvatures.
The paper generalizes structures for groups from curved spaces.
problem Generalizing Z-structures and EZ-structures to curved spaces. method Analyzing fundamental groups of curved spaces and applying Z-structures and EZ-structures. result Fundamental groups of curved spaces admit Z-structures and EZ-structures. Procedure groups nonparametric regression curves automatically.
problem Determining groups of nonparametric regression curves when curves are numerous.
method Automatic selection of group number through testing procedure.
result Groups of nonparametric regression curves exist in tunnel geometry.
Paper finds infinite pairs of fiber-type curves with same topology but different embeddings.
problem Conditions for curves in projective surfaces to have specific fundamental groups.
method Examine fiber-type curves in P2 and use twisted Alexander polynomials. result Infinite Zariski pairs of fiber-type curves with non-isomorphic fundamental groups.
Study quotients of curve complex actions by mapping class group.
problem Understanding actions of mapping class group on curve complex quotients.
method Cone off uniformly quasi-convex subspaces to form symmetric curve sets, non-maximal train track sets, and compression body disc sets. Analyze actions of mapping class group on these quotients.
result Actions of mapping class group on quotients are strongly WPD, non-elementary, and have infinite diameter.
Study shows surfaces without certain curves have infinite orbit graph.
problem Characterizing surfaces with specific curve properties.
method Utilized tools from mapping class group geometry.
result Infinite-invariance index 1 surfaces lack good curve graphs.
Three geometric analysis results on curve flows and Lie groups.
problem Analyzing geometric flows and Lie groups.
method Curve-shortening flow, point-wise curvature preserving flow, Lie group analysis.
result Interpolation between Sol and hyperbolic space in Lie groups.
Automorphisms of fine curve graphs match surface homeomorphisms for planar surfaces.
problem Understanding automorphisms of fine curve graphs on surfaces.
method Analyzing vertices and edges of fine curve graphs to match with surface homeomorphisms.
result Automorphism group of fine curve graphs is naturally isomorphic to the homeomorphism group of boundaryless planar surfaces with at least 7 punctures.
In this short note we survey some results about the fundamental group of a compact negatively curved manifold. In particular, we review a theorem of Gusevskij, it states that the fundamental group of a compact negatively curved manifold does not belong to C, where C is the smallest class of grou…
Develops intrinsic curved cosets for Cartan geometries.
problem Defines curved cosets for arbitrary Cartan geometries.
method Defines intrinsic holonomy group and curved cosets.
result Curved cosets retain characteristics of homogeneous counterparts and behave well under automorphisms.
Mapping class group subgroups yield quasi-isometric curve complex.
problem Understanding the curve complex through coset intersections.
method Proving quasi-isometry and combinatorial equivalence of curve complex and coset intersection complex.
result Automorphism group of coset intersection complex is the extended mapping class group.
In this paper, we adapt the differential signature construction to the equivalence problem for complex plane algebraic curves under the actions of the projective group and its subgroups. Given an action of a group G, a signature map assigns to a plane algebraic curve another plane algebraic curve (a signature curve) …
Groups with specific curvature have a regular language of geodesics.
problem Understanding the language of geodesics in non-positively curved triangle groups.
method Proving finitely many cone types and regularity of geodesic languages.
result The language of lexicographically first geodesics is regular and satisfies the fellow traveller property.
We characterize the biharmonic curves in the special linear group SL(2,R). In particular, we show that all proper biharmonic curves in SL(2,R)are helices and we give their explicit parametrizations as curves in the pseudo-Euclidean space R24.
The geometric monodromy of a plane curve singularity is a quasi-finite diffeomorphism. In this paper we locate the reduction curves of the geometric monodromy and the quadratic vanishing cycles of the singularity. An application to the geometric monodromy group is given.
Study of orbifold mapping class groups via arc and curve actions.
problem Understanding the structure of orbifold mapping class groups.
method Defined orbifold mapping class groups and studied their actions on arcs and curves. Established a Birman exact sequence and derived finite presentations.
result Finite presentations of orbifold mapping class groups established.
In this paper, new representations of a Bertrand curve pair in three dimensional Lie groups with bi-invariant metric are given. Besides, the spherical indicatrices of a Bertrand curve pair are obtain and the relations between the spherical indicatrices and new representations of Bertrand curve pair are shown.
In this note, we present a new method for computing fundamental groups of curve complements using a variation of the Zariski-Van Kampen method on general ruled surfaces. As an application we give an alternative (computation-free) proof for the fundamental group of generic (p,q)-torus curves.
Minimal surfaces in Heisenberg group have null curves and lines.
problem Characterizing timelike minimal surfaces in the Heisenberg group.
method Characterization through null curves and lines with prescribed curvatures.
result Minimal surfaces are defined by the multiplication of null curves and affine null lines.
Machine learning predicts Shafarevich-Tate group orders of elliptic curves.
problem Predicting the order of the Shafarevich-Tate group of elliptic curves.
method Train feed-forward neural network and regression models on elliptic curve invariants.
result Models achieve high accuracy (>0.9) and predict orders not seen during training. We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.