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…
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48 results for “arithmetic hyperbolic 3-orbifolds”
The paper shows that arithmetic hyperbolic 3-orbifolds have many non-commensurable pairs with similar geodesic spectra.
problem Understanding the relationship between geodesic length spectra and commensurability of arithmetic hyperbolic 3-orbifolds.
method Using a bounded gaps result for prime ideals in number fields, the paper constructs infinitely many non-commensurable pairs of arithmetic hyperbolic 3-orbifolds with similar geodesic spectra.
result Arithmetic hyperbolic 3-orbifolds can have many non-commensurable pairs with similar geodesic spectra.
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.
Study shows Betti numbers of curves and orbifolds relate to volume and genus.
problem Understanding Betti numbers of Shimura curves and 3-orbifolds.
method Analyzes asymptotic behavior of Betti numbers in relation to volume and genus.
result Gauss-Bonnet equality for Shimura curves and vanishing Betti numbers for 3-orbifolds.
Our main result is that for all sufficiently large , the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field and systole bounded below by has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds wit…
Study on geodesic surfaces in hyperbolic 3-orbifolds, proving overlaps in area sets.
problem Understanding overlaps in geometric genus spectra of non-commensurable hyperbolic 3-orbifolds.
method Defined geometric genus spectrum and totally geodesic area set, proving results on overlaps.
result Arithmetic hyperbolic 3-orbifolds can have large overlaps in their totally geodesic area sets.
Study on counting Salem numbers linked to geodesics in hyperbolic orbifolds.
problem Quantifying Salem numbers associated with closed geodesics in arithmetic hyperbolic orbifolds.
method Analytical and asymptotic methods to estimate the number of square-rootable Salem numbers.
result Found that non-compact arithmetic 3-dimensional orbifolds define square-rootable Salem numbers of degree 4.
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.
Let G be an arithmetic Kleinian group, and let O be the associated hyperbolic 3-orbifold or 3-manifold. In this paper, we prove that, in many cases, G is large, which means that some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. This has many consequences, including that O has in…
Let O be a compact orientable 3-orbifold with non-empty singular locus and a finite volume hyperbolic structure. (Equivalently, O is the quotient of hyperbolic 3-space by a lattice in PSL(2,C) with torsion.) Then we prove that O has a tower of finite-sheeted covers {O_i} with linear growth of p-homology, for some prime…
Study on projective structures on a hyperbolic 3-orbifold using tetrahedra.
problem Analyzing projective structures on hyperbolic 3-orbifolds.
method Parameterization using classical invariants, traces, and geometric cross ratios.
result Computed and analyzed the moduli space of projective structures.
Authors create non-isometric 3-orbifolds with identical topology and volume.
problem Finding non-isometric hyperbolic 3-orbifolds with the same topological type and volume.
method Constructing pairs of non-isometric hyperbolic 3-orbifolds with the same topological type and volume.
result Demonstrated the existence of non-isometric hyperbolic 3-orbifolds with the same topological type and volume.
Study spectral gaps and bass notes of random hyperbolic 3-orbifolds.
problem Investigate spectral properties of random hyperbolic 3-orbifolds.
method Analyze two models of random hyperbolic 3-orbifolds related to Apollonian and super Apollonian groups.
result Explicit spectral gaps determined for random orbifolds.
Study projective deformations of hyperbolic 3-orbifolds with turnover ends.
problem Deformations of hyperbolic 3-orbifolds with turnover ends in projective geometry.
method Projective deformations of hyperbolic 3-orbifolds with turnover ends, focusing on totally geodesic generalized cusps.
result Turnover funnels remain totally geodesic and the deformed projective 3-orbifold remains properly convex.
We construct the first example of a ``one-cusped'' hyperbolic 3-orbifold for which we see the true shape of the space of hyperbolic Dehn fillings.
Researchers found a new hyperbolic 3-orbifold using a Menger curve.
problem Constructing a new hyperbolic 3-orbifold with specific properties.
method Discovered a discrete, convex cocompact and faithful representation of a hyperbolic group into PU(2,1).
result The 3-orbifold at infinity of the representation is a closed hyperbolic 3-orbifold.
Study finds volume lower bounds for specific 3-orbifolds.
problem Finding volume lower bounds for hyperbolic 3-orbifolds with certain suborbifolds.
method Analyzing the topology of essential 2-suborbifolds and computing their guts.
result Obtained lower bounds on the volume of specific hyperbolic 3-orbifolds.
By using Klein's model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev's theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real p…
We show that any immersion, which is not a covering of an embedded 2-orbifold, of a totally geodesic hyperbolic turnover in a complete orientable hyperbolic 3-orbifold is contained in a hyperbolic 3-suborbifold with totally geodesic boundary, called the "turnover core,'' whose volume is bounded from above by a function…
Neumann and Reid described in their paper "Rigidity of cusps in deformations of hyperbolic 3-orbifolds" (Math Ann. 295 (1993) no. 2, 223--237) a 2-cusped hyperbolic 3-orbifold in which the cusps are geometrically isolated. Based on numerical evidence provided by Jeff Weeks' snappea program, they conjectured that the cu…
Study on volume and homology of hyperbolic 3-orbifolds, proving bounds on first homology group.
problem Volume and homology of hyperbolic 3-orbifolds, focusing on irreducible cases.
method Application of results from sequel, proving bounds without irreducibility assumption.
result Bounds on first homology group dimensions for hyperbolic 3-orbifolds.
For each natural number n >= 4, we determine the unique lowest volume hyperbolic 3-orbifold whose torsion orders are bounded below by n. This lowest volume orbifold has base space the 3-sphere and singular locus the figure-8 knot, marked n. We apply this result to give sharp lower bounds on the volume of a hyperbolic m…
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.
Finite subgroups of good groups correspond to their profinite completions.
problem Characterizing finite subgroups of profinite completions of good groups.
method Proving bijective correspondence between conjugacy classes of finite p-subgroups in and , and analyzing centralizers and normalizers.
result Bijective correspondence between conjugacy classes of finite p-subgroups in and .
We prove that a closed 3-orbifold that fibers over a hyperbolic polygonal 2-orbifold admits a family of hyperbolic cone structures that are viewed as regeneration of the polygon, provided that the perimeter is minimal.
Geodesically embeds simplest arithmetic hyperbolic manifolds into higher dimensions.
problem Embedding arithmetic hyperbolic manifolds.
method Proving embedding into arithmetic hyperbolic -manifolds or their universal Abelian covers.
result Arithmetic hyperbolic manifolds of simplest type can be geodesically embedded.
New method shows how certain groups act on 3-orbifolds.
problem Understanding how groups act on 3-dimensional spaces.
method Using veering pairs of laminations and loom spaces.
result Groups with invariant veering pairs are hyperbolic 3-orbifold groups.
Study upper bounds on orbifold volume and its homology.
problem Bounding the volume and homology of hyperbolic 3-orbifolds.
method Analyzes the relationship between orbifold volume and its first homology group's dimension over the field of two elements.
result Strict upper bounds on orbifold volume imply upper bounds on its first homology group's dimension.
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.
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.
Arithmetic 3-manifolds with infinite geodesics are proven.
problem Arithmeticity of 3-manifolds with infinitely many geodesics.
method Analysis of totally geodesic surfaces in hyperbolic 3-manifolds.
result Arithmetic 3-manifolds with infinite geodesics are proven.
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].
Torsion in cohomology grows exponentially for certain arithmetic hyperbolic manifolds.
problem Growth of torsion in cohomology of arithmetic hyperbolic manifolds.
method Study of sequences of arithmetic subgroups of SO_0(d,1) and Spin(d,1) yielding hyperbolic manifolds.
result Torsion in cohomology grows exponentially under natural assumptions.
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.
The paper proves many hyperbolic manifolds have similar growth rates to all hyperbolic manifolds.
problem Understanding growth rates of hyperbolic manifolds.
method Proving existence of many compact hyperbolic manifolds as boundaries of other hyperbolic manifolds.
result Establishes that compact hyperbolic arithmetic n-manifolds have the same growth rate as all hyperbolic n-manifolds.
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.
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.
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.
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…
The heat coefficients related to the Laplace-Beltrami operator defined on the hyperbolic compact manifold $H^3/\Ga$ are evaluated in the case in which the discrete group $\Ga$ contains elliptic and hyperbolic elements. It is shown that while hyperbolic elements give only exponentially vanishing corrections to the trace…
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.