New classification of hyperbolic Coxeter prisms.
problem Classifying hyperbolic Coxeter prisms.
method Determine which prisms are quasi-arithmetic or arithmetic.
result New insights into commensurability and systoles of associated orbifolds.
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48 results for “arithmetic hyperbolic”
The study of systoles in arithmetic hyperbolic manifolds.
problem Understanding the systoles of arithmetic hyperbolic manifolds.
method Construction and analysis of arithmetic hyperbolic manifolds.
result Explicit bounds on volumes and systoles of arithmetic hyperbolic manifolds.
The paper explores subspaces in hyperbolic lattices and their arithmetic properties.
problem Arithmeticity criterion for hyperbolic lattices and suborbifolds.
method Analysis of totally geodesic suborbifolds and Vinberg's commensurability invariants.
result Arithmeticity of hyperbolic orbifolds is linked to the existence of infinitely many fc-subspaces.
We prove that any arithmetic hyperbolic -manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic -manifold or its universal Abelian cover can.
Study general hyperbolic gluings, proving quasi-arithmeticity of building blocks.
problem Proving quasi-arithmeticity of building blocks in hyperbolic gluings.
method Generalized gluings of hyperbolic orbifolds, proving quasi-arithmeticity.
result Building blocks of quasi-arithmetic gluings must also be quasi-arithmetic.
Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.
problem Determining when a flat manifold can be a cusp cross-section in arithmetic hyperbolic manifolds.
method Analyzing rational representations of holonomy groups and quasi-arithmetic manifolds.
result Conditions for a flat manifold to appear as a cusp cross-section in every commensurability class of arithmetic hyperbolic manifolds.
New method constructs non-arithmetic hyperbolic orbifolds from complex arithmetic ball quotients.
problem Creating non-arithmetic lattices in projective orthogonal groups.
method Using anti-holomorphic involutions on complex arithmetic ball quotients, gluing fixed loci along geodesic subspaces.
result Explicit calculation of the volume of constructed non-arithmetic orbifolds.
Study finds limit points of bass notes on hyperbolic surfaces.
problem Understanding the limit points of bass notes on hyperbolic surfaces.
method Proved using mathematical analysis of arithmetic hyperbolic surfaces.
result Limit points are found to be within the interval [0, 1/4].
Geometric constraints help classify hyperbolic polytopes.
problem Classifying reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.
method Established geometric constraints on compact Coxeter polytopes in hyperbolic spaces.
result Geometric constraints are useful for classifying hyperbolic polytopes.
The paper embeds non-arithmetic hyperbolic manifolds into higher-dimensional spaces.
problem Embedding non-arithmetic hyperbolic manifolds into higher-dimensional hyperbolic spaces.
method Using totally geodesic submanifolds and commensurability classes.
result Many non-arithmetic hyperbolic manifolds can be embedded geodesically.
Bound on diameter of arithmetic hyperbolic orbifolds.
problem Estimating the maximum distance between points in arithmetic hyperbolic orbifolds.
method Using a specific formula involving logarithm of volume and Cheeger constant.
result Diameter of arithmetic hyperbolic orbifolds is bounded by a function of volume and Cheeger constant.
Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
problem Identifying Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
method Comprehensive classification of commensurability classes of cusped, arithmetic, and non-arithmetic complex hyperbolic 2-manifolds.
result Some Nil 3-manifolds are cross-sections in every commensurability class, while others are cross-sections in only one.
Paper finds new 3D shapes that can be inside a 4D space.
problem Finding new 3D shapes with specific properties.
method Examined arithmetic hyperbolic 3-manifolds and their homology.
result Discovered infinitely many 3D shapes that are rational homology spheres and can bound geometrically.
New geometric invariant limits the number of semi-arithmetic groups.
problem Understanding the structure of semi-arithmetic Fuchsian groups.
method Introducing a new geometric invariant called stretch and using the arithmetic Margulis lemma.
result There exist only finitely many conjugacy classes of semi-arithmetic groups with bounded arithmetic dimension, stretch, and coarea.
The paper improves Vinberg's algorithm for arithmetic hyperbolic lattices.
problem Finding maximal reflection sublattices in arithmetic hyperbolic lattices.
method Provided an effective termination condition for Vinberg's semi-algorithm.
result The algorithm becomes an effective method for finding maximal reflection sublattices.
The study identifies flat manifolds with unique cusp cross-sections in arithmetic hyperbolic manifolds.
problem Characterizing flat manifolds that have unique cusp cross-sections in arithmetic hyperbolic manifolds.
method Algebraic characterization of cusp cross-sections in arithmetic hyperbolic manifolds.
result Construction of flat manifolds with unique cusp cross-sections and proof of their existence in all dimensions .
We will show that, for any noncompact arithmetic hyperbolic -manifold with , and any compact arithmetic hyperbolic -manifold with that is not a -dimensional arithmetic hyperbolic manifold defined by octonions, its fundamental group is not LERF. The main ingredient in the proof is a study on abelia…
We show that closed arithmetic hyperbolic n-dimensional orbifolds with larger and larger volumes give rise to triangulations of the underlying spaces whose 1-skeletons are harder and harder to embed nicely in Euclidean space. To show this we generalize an inequality of Gromov and Guth to hyperbolic n-orbifolds and find…
The study finds that certain hyperbolic manifolds contain subgroups isomorphic to surface groups.
problem The existence of thin surface subgroups in non-uniform arithmetic lattices.
method Analyzes arithmetic hyperbolic manifolds and their fundamental groups.
result Fundamental groups of non-compact arithmetic hyperbolic manifolds contain thin surface subgroups.
In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of tech…
Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing ground fields of arithmetic hyperbolic reflection groups are defined, and good bounds of their degrees (over Q) are obtained. For example, degree of the ground field of any arithmetic hyperbolic reflection group in dimension at…
In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently construct…
We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.
Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifoldsmath.DS
Complex hyperbolic manifolds with many totally geodesic submanifolds are arithmetic.
problem Characterizing arithmeticity of complex hyperbolic manifolds with certain submanifolds.
method Developing superrigidity theorems for complex hyperbolic lattices and proving nonexistence of certain maps.
result Finite volume complex hyperbolic -manifolds containing infinitely many maximal totally geodesic submanifolds of dimension at least two are arithmetic.
Minimal non-arithmetic hyperbolic 3-orbifold found with least volume.
problem Finding the hyperbolic 3-orbifold with minimal volume among non-arithmetic ones.
method Utilized the tetrahedral Coxeter group and horoball configuration to prove minimal volume.
result The 1-cusped quotient of hyperbolic space by the tetrahedral Coxeter group has minimal volume.
This paper continues arXiv.org:math.AG/0609256, arXiv:0708.3991 and arXiv:0710.0162 . Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimension at least 3 are defined, and explicit bounds of their degrees (over …
Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about v^v such manifolds of volume at most v, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, alm…
It is a longstanding problem to determine the precise relationship between the geodesic length spectrum of a hyperbolic manifold and its commensurability class. A well known result of Reid, for instance, shows that the geodesic length spectrum of an arithmetic hyperbolic surface determines the surface's commensurabilit…
Systoles of hyperbolic manifolds are dense and related to Salem numbers.
problem Density of systoles in hyperbolic manifolds.
method Analyzing systoles and arithmetic hyperbolic manifolds.
result Systoles of closed arithmetic hyperbolic manifolds are dense in .
Study shows spectral gaps limit points on surfaces.
problem Understanding spectral gaps on arithmetic hyperbolic surfaces.
method Analyzes closed arithmetic hyperbolic surfaces to find limit points of spectral gaps.
result Limit points of spectral gaps are on the interval [0, 1/4].
The purpose of the present paper is to prove existence of super-exponentially many compact orientable hyperbolic arithmetic -manifolds that are geometric boundaries of compact orientable hyperbolic -manifolds, for any , thereby establishing that these classes of manifolds have the same growth rate w…
We prove, under the assumption of the virtual fibration conjecture for arithmetic hyperbolic 3-manifolds, that all arithmetic lattices in O(n,1), n> 4, and different from 7, are non-coherent. We also establish noncoherence of uniform arithmetic lattices of the simplest type in SU(n,1), n> 1, and of uniform lattices in …
Explains how arithmetic manifolds solve geometric questions about systole and kissing number.
problem Geometric questions about systole and kissing number in hyperbolic manifolds.
method Use of arithmetic manifolds to solve geometric questions.
result Answers geometric questions about systole and kissing number for dimension 2 and higher dimensions.
New research shows certain arithmetic lattices can't be LERF.
problem Determining if arithmetic lattices are LERF.
method Analyzing trialitarian arithmetic lattices in .
result Trialitarian arithmetic lattices in are not LERF.
Minimal crossing number found in arithmetic curve systems.
problem Finding the minimal crossing number in arithmetic curve systems.
method Analyzing systoles of hyperbolic surfaces associated with congruence lattices in SL2(Z).
result Minimal crossing number is asymptotically achieved.
We determine the minimal volume of arithmetic hyperbolic orientable n-dimensional orbifolds (compact and non-compact) for every odd dimension n>3. Combined with the previously known results it solves the minimal volume problem for arithmetic hyperbolic n-orbifolds in all dimensions.
We prove that if a closed hyperbolic 3-manifold M contains infinitely many totally geodesic surfaces, then M is arithmetic.
New manifolds with small systoles not quasi-arithmetic.
problem Finding manifolds with small systoles not quasi-arithmetic.
method Hybrid construction of known manifolds.
result Exhibited manifolds with arbitrarily small systoles.
We determine the three hyperbolic 5-orbifolds of smallest volume among compact arithmetic orbifolds, and we identify their fundamental groups with hyperbolic Coxeter groups. This gives two different ways to compute the volume of these orbifolds.
The study finds infinite commensurability classes of hyperbolic manifolds with Salem number lengths.
problem Understanding commensurability classes of arithmetic hyperbolic manifolds.
method Analyzing Salem numbers and their relation to arithmetic hyperbolic manifolds.
result Infinitely many commensurability classes of arithmetic hyperbolic manifolds with geodesic lengths equal to logarithms of Salem numbers.
In this paper we prove that there is a direct relationship between Salem numbers and translation lengths of hyperbolic elements of arithmetic hyperbolic groups that are determined by a quadratic form over a totally real number field. As an application we determine a sharp lower bound for the length of a closed geodesic…
Study on curves on specific arithmetic quotients of hyperbolic 2-ball.
problem No complex curves of certain genus on these arithmetic quotients.
method Volume estimates and understanding special subvarieties.
result For large discriminants, no complex curves of fixed genus.
New groups found in hyperbolic space with infinite fields of definition.
problem Finding maximal reflection groups in hyperbolic space.
method Developed new quasi-arithmetic reflection groups for hyperbolic 2-space.
result Infinitely many maximal quasi-arithmetic reflection groups with unbounded field degrees.
Classifies non-arithmetic orbifolds in specific hyperbolic spaces.
problem Classifying non-arithmetic affine invariant orbifolds in Hodd(2, 2) and H(3, 1).
method Classification through Veech surfaces and rigidity results.
result Classification of non-arithmetic rank one orbifolds.
We discuss the geometry of some arithmetic orbifolds locally isometric to a product of real hyperbolic spaces of dimension two and three, and prove that certain sequences of non-uniform orbifolds are convergent to this space in a geometric ("Benjamini--Schramm") sense for hyperbolic three--space and a product of hyperb…
Study on hyperbolic polyhedra and their volume, proving finiteness of arithmetic groups.
problem Finiteness of arithmetic maximal reflection groups in hyperbolic polyhedra.
method Observation of volume distribution and recent work with M. Fraczyk and S. Hurtado.
result Proof of finiteness of arithmetic maximal reflection groups.
A hyperbolic reflection group is a discrete group generated by reflections in the faces of an -dimensional hyperbolic polyhedron. This survey article is dedicated to the study of arithmetic hyperbolic reflection groups with an emphasis on the results that were obtained in the last ten years and on the open problems.
Study finds bounds for systole length on arithmetic punctured spheres.
problem Finding the shortest essential curve on arithmetic punctured spheres.
method Correspondence between surfaces and planar triangulations to bound systole length.
result Arithmetic surfaces do not achieve maximal systole length for .