Shorter sides in geodesic triangles in hyperbolic plane.
problem Properties of geodesic triangles in hyperbolic surfaces.
method Analyzing lifts of a closed geodesic in hyperbolic 2-space.
result Sides of triangles formed by geodesics are shorter than the geodesic itself.
Complex hyperbolic triangle groups were first considered by Mostow in building the first nonarithmetic lattices in PU(2, 1). They are a natural generalization of the classical triangle groups acting on the hyperbolic plane. A well-known theorem of Takeuchi is that there are only finitely many Fuchsian triangle groups t…
Napoleonic triangles don't exist in hyperbolic geometry.
problem The existence of Napoleonic triangles in hyperbolic geometry.
method Analyzing the construction of equilateral triangles on hyperbolic triangles.
result Hyperbolic triangles do not form Napoleonic triangles, except equilateral ones.
Classifies complex hyperbolic triangle groups by types.
problem Classifying complex hyperbolic triangle groups.
method By types defined by the ellipticity of two short words.
result Improves Schwartz conjecture.
We classify the 3-dimensional hyperbolic polyhedral orbifolds that contain no embedded essential 2-suborbifolds, up to decomposition along embedded hyperbolic triangle orbifolds (turnovers). We give a necessary condition for a 3-dimensional hyperbolic polyhedral orbifold to contain an immersed (singular) hyperbolic tur…
Study stabilizers of complex hyperbolic triangle groups, finding generators and signatures.
problem Understanding the stabilizers of complex hyperbolic triangle groups.
method Explicit generators and signatures of stabilizers computed for each group orbit of mirrors.
result Explicit generators and signatures of stabilizers for some triangle groups.
Complex Hyperbolic Triangle Groups of Type [m,m,0;3,3,2]math.GT Study on discrete complex hyperbolic triangle groups of specific type.
problem Discreteness of complex hyperbolic triangle groups of type [m,m,0;3,3,2]. method Analysis of groups generated by complex reflections with specific orders and distances.
result Determined intervals in parameter space for discrete and non-discrete groups.
Triangle groups show rigidity in hyperbolic spaces.
problem Local rigidity of triangle groups generated by reflections.
method Geometric representation and diagonal embeddings in PGL(2,R) and PSp±(2n,R). result Triangle groups are locally rigid in hyperbolic spaces.
For k>6, we determine the minimal area of a compact hyperbolic surface, and an oriented compact hyperbolic surface that can be tiled by embedded regular triangles of angle 2π/k. Based on this, all the cases of equality in Laszlo Fejes Toth's triangle bound for hyperbolic surfaces are described.
Complex Hyperbolic Triangle Groups of Type [m,m,0;n1,n2,2]math.GT The study determines discreteness of complex hyperbolic triangle groups.
problem Discreteness of complex hyperbolic triangle groups of specific type.
method Analysis of groups generated by complex reflections with given orders and distances.
result Determined intervals in parameter space for discrete and non-discrete groups.
New complex hyperbolic lattices discovered from triangle groups.
problem Finding new non-arithmetic complex hyperbolic lattices.
method General procedure to produce fundamental domains for complex hyperbolic triangle groups.
result Some triangle groups yield new commensurability classes, increasing the count to 22.
Study of subgroups in complex hyperbolic lattice triangle groups.
problem Characterizing subgroups of finite index in complex hyperbolic lattice triangle groups.
method Explicit construction and analysis of subgroups, examination of their properties.
result Identification of neat subgroups, subgroups with positive first Betti number, and homomorphisms onto non-Abelian free groups.
Study on discrete properties of complex hyperbolic triangle groups.
problem Discreteness of complex hyperbolic triangle groups of type [m1, m2, 0].
method Analysis of isometries generated by complex reflections in ultra-parallel geodesics.
result Proves discreteness and non-discreteness results for these groups.
Random hyperbolic surfaces have low Cheeger constants.
problem Estimating Cheeger constants of random hyperbolic surfaces.
method Modeling random hyperbolic surfaces using ideal triangles and analyzing their Cheeger constants.
result Generic hyperbolic surfaces have Cheeger constants less than 3/2π + ε.
Study classifies 4 types of 2-fold symmetric complex hyperbolic triangle groups.
problem Classifying 2-fold symmetric complex hyperbolic triangle groups.
method Examined groups generated by reflections through angle 2pi/p, focusing on elliptic elements.
result Found only 4 types of groups that could be discrete.
Complex hyperbolic triangle groups are discrete when certain conditions are met.
problem Proving discreteness of complex hyperbolic triangle groups.
method Analyzing representations and isometries to determine discreteness.
result Conditions for discreteness of complex hyperbolic triangle groups are identified.
In this paper we mainly pay attention to the complex hyperbolic triangle groups of type (m, n, infinity) and discuss the discreteness. From the results more explicit conclusions about the triangle groups of type (n, infinity, infinity) will also be given.
In this paper we describe trigonometry on the de Sitter surface. For that a characterization of geodesics is given, leading to various types of triangles. We define lengths and angles of these. Then, transferring the concept of polar triangles from spherical geometry into the Minkowski space, we relate hyperbolic with …
Complex hyperbolic triangle groups yield specific 3-manifolds at infinity.
problem Characterizing manifolds at infinity of complex hyperbolic orbifolds.
method Spherical CR uniformization and Dehn surgery.
result Specific 3-manifolds at infinity of complex hyperbolic triangle groups are identified.
Identifies Anosov representations of hyperbolic triangle groups in SL(3,R).
problem Classifying Anosov representations of hyperbolic triangle groups into SL(3,R).
method Proving representations are Anosov if they lie in the Hitchin component or the Barbot component, with specific conditions for eigenvalues.
result Anosov representations in SL(3,R) have non-convex boundary maps.
New theorem disproves Angle Defect for super triangles.
problem Angle Defect Theorem for N=1 super hyperbolic geometry.
method Action of OSp(1|2) on real super Minkowski space and brute-force computation.
result Disproves Angle Defect Theorem and provides novel additive function.
Paper shows minimum 10 vertices for hyperbolic origami 2-torus.
problem Finding minimum vertices for hyperbolic origami 2-torus.
method Geodesic triangulation and isometric polyhedral embedding.
result 10 vertices are the minimum required for a hyperbolic origami 2-torus.
We prove, in the context of Hilbert geometry, the equivalence between the existence of an upper bound on the area of ideal triangles and the Gromov-hyperbolicity.
Study on hyperbolic triangles and once-punctured torus groups, focusing on group relations and deformations.
problem Understanding group relations and deformations in hyperbolic geometry.
method Analyzing the deformation space of singular hyperbolic metrics on a torus and studying the holonomy map.
result For most hyperbolic triangle areas, the group generated by rotations has no nontrivial relations, while for some, it does.
We consider the Neumann Laplacian acting on square-integrable functions on a triangle in the hyperbolic plane that has one cusp. We show that the generic such triangle has no eigenvalues embedded in its continuous spectrum. To prove this result we study the behavior of the real-analytic eigenvalue branches of a degener…
We present several formulas for the traces of elements in complex hyperbolic triangle groups generated by complex reflections. The space of such groups of fixed signature is of real dimension one. We parameterise this space by a real invariant alpha of triangles in the complex hyperbolic plane. The main result of the p…
Paper connects knot theory with cluster algebra via Alexander polynomials.
problem Understanding Alexander polynomials for 2-bridge knots.
method Use of cluster variables and ancestral triangles.
result Alexander polynomials are specializations of cluster variables.
Art and science of conformally correct tilings of compact surfaces.
problem Producing conformally correct tilings of compact surfaces.
method Discussing and presenting examples of tilings.
result Presentation of a tiling of the Chmutov surface by hyperbolic triangles.
3-manifolds are CR uniformized on spheres, proving a conjecture.
problem Uniformizing 3-manifolds with cusps using CR methods.
method Spherical CR uniformization of complex hyperbolic triangle groups.
result Magic 3-manifolds and other cusped 3-manifolds are CR uniformizable.
The study defines fields of definition for triangle groups as Fuchsian groups.
problem Characterizing the fields of definition for triangle groups as Fuchsian groups.
method Analyzing the trace field and properties of compact hyperbolic triangle groups.
result Exactly eleven compact hyperbolic triangle groups are conjugate to subgroups of \(\mathrm{PSL}_2(K)\) where \(K\) is a specific field.
We prove a conjecture of R. Schwartz about the type of some complex hyperbolic triangle groups.
The theory of complex hyperbolic discrete groups is still in its childhood but promises to grow into a rich subfield of geometry. In this paper I will discuss some recent progress that has been made on complex hyperbolic deformations of the modular group and, more generally, triangle groups. These are some of the simpl…
The study proves conjecture for specific Artin groups.
problem Proving conjecture about Artin groups' properties.
method Analyzing Artin groups associated to triangle-free graphs and cones over square-free bipartite graphs.
result Proves conjecture for specific Artin groups.
In this note we prove that a complex hyperbolic triangle group of type (m,m,infinity), i.e. a group of isometries of the complex hyperbolic plane, generated by complex reflections in three complex geodesics meeting at angles Pi/m, Pi/m and 0, is not discrete if the product of the three generators is regular elliptic.
Study Steklov eigenvalues on hyperbolic triangle-tiling graphs.
problem Analyzing Steklov eigenvalues on specific hyperbolic graph structures.
method Introduced a graph roughly isometric to hyperbolic plane, used discretization to transfer bounds.
result Steklov eigenvalues tend to zero proportionally to the inverse of the domain size.
Maps between acute triangles with minimal stretch found and studied.
problem Finding the minimal stretch between acute triangles.
method Formula for the smallest Lipschitz constant and analysis of the metric space.
result Metric space of pairs of acute triangles with fixed area is Finsler and geodesics determined.
In this paper we study the area of ideals triangles in a convex domain with its Hilbert geometry. We obtain a characterization of the hyperbolic geometry among all the Hilbert geometry in terms of area of ideals triangles. We also obtain a sharp lower bound on the hilbert area of ideal triangles, independant of the con…
New dynamical approach defines symmedian as hyperbolic barycenter.
problem Understanding symmedian properties in hyperbolic geometry.
method Developed a new dynamical coordinatization.
result Symmedian point acts as hyperbolic barycenter.
The paper classifies discrete complex hyperbolic triangle groups.
problem Classifying discrete complex hyperbolic triangle groups.
method Analyzing complex hyperbolic spaces and isometries.
result Classifies discrete complex hyperbolic (n,∞,∞)-triangle groups for n=3,4,5. We determine the lowest volume hyperbolic Coxeter polyhedron whose corresponding hyperbolic polyhedral 3-orbifold contains an essential 2-suborbifold, up to a canonical decomposition along essential hyperbolic triangle 2-suborbifolds.
New CR representations are found and shown to be redundant.
problem Identifying and classifying CR representations of 3-manifolds.
method Experimental computation of limit sets and exact computations of triangle groups.
result Many CR representations are redundant and conjugate.
Study shortest non-simple geodesics on 2-orbifolds, finding unique shortest curve.
problem Finding shortest non-simple closed geodesics disjoint from orbifold points.
method Fundamental domains and hyperbolic trigonometry.
result Identified and classified all figure eight geodesics on triangle group orbifolds.
Let M be either the 2-sphere $\SS^2 \subset\RR^3$ or the hyperbolic plane $\HH^2 \subset \RR^3$. If Δ(abc) is a geodesic triangle on M with corners at a,b,c∈M, we denote by α,β,γ∈M the midpoints of their sides. If Ω denotes the oriented area of this triangle on M, it satisfies the relations: $$ \s…
Three counterexamples show higher eigenvalue multiplicities than conjectured.
problem Determining the maximum eigenvalue multiplicity for closed hyperbolic surfaces.
method Applying the twisted Selberg trace formula to induced representations of triangle groups.
result Found counterexamples with higher eigenvalue multiplicities than previously conjectured.
The paper generalizes Farey tessellation to 3D hyperbolic space.
problem Generalizing Farey tessellation to higher dimensions.
method Introducing conformal bryophylla and classifying them.
result Properties of conformal bryophylla's limiting sets studied.
The study introduces hyperbolic angles in Lorentzian spaces and characterizes curvature bounds.
problem Characterizing timelike curvature bounds in Lorentzian spaces.
method Synthetic geometric framework of Lorentzian (pre-)length spaces, introduction of hyperbolic angles, and angle monotonicity condition.
result Characterization of timelike curvature bounds with an angle monotonicity condition.
Let X be a geodesic metric space. Gromov proved that there exists k>0 such that if every sufficiently large triangle T satisfies the Rips condition with constant k times pr(T), where pr(T) is the perimeter T, then X is hyperbolic. We give an elementary proof of this fact, also giving an estimate for k. We also show tha…
Few years ago we developed jointly with I.Dynnikov new discretization of complex analysis (DCA) based on the two-dimensional manifolds with colored black/white triangulation. Especially deep results were obtained for the Euclidean plane with equilateral triangle lattice. In the present work we develop a DCA theory for …