Hyperbolic N-space Research Articles

Local behavior of the Eden model on graphs and tessellations of manifolds

The Eden Model in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document} constructs a blob as follows: initially a single unit hypercube is infected, and each second a hypercube adjacent to the infected ones is selected randomly and infected. Manin, Roldán, and Schweinhart investigated the topology of the Eden model in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^{n}$$\\end{document} by considering the possible shapes which can appear on the boundary. In particular, they give probabilistic lower bounds on the Betti numbers of the Eden model. In this paper, we prove analogous results for the Eden model on any infinite, vertex-transitive, locally finite graph: with high probability as time goes to infinity, every “possible” subgraph (with mild conditions on what “possible” means) occurs on the boundary of the Eden model at least a number of times proportional to an isoperimetric profile of the graph. Using this, we can extend the results about the topology of the Eden model to non-Euclidean spaces, such as hyperbolic n-space and universal covers of certain Riemannian manifolds.

New lower bounds for optimal horoball packing density in hyperbolic n-space for 6 le n le 9

Koszul type Coxeter simplex tilings exist in hyperbolic n-space mathbb {H}^n up to n = 9, and their horoball packings achieve the highest known regular ball packing densities for n = 3, 4, 5. In this paper we determine the optimal horoball packing densities of Koszul simplex tilings in dimensions 6 le n le 9, which give new lower bounds for optimal packing density in each dimension. The symmetries of the packings are given by Coxeter simplex groups, and a parameter related to the Busemann function gives an isometry invariant description of different optimal horoball packing configurations.

Effective bounds for Vinberg’s algorithm for arithmetic hyperbolic lattices

A group of isometries of a hyperbolic n-space is called a reflection group if it is generated by reflections in hyperbolic hyperplanes. Vinberg gave a semi-algorithm for finding a maximal reflection sublattice in a given arithmetic subgroup of $${{\,\textrm{O}\,}}(n,1)$$ of the simplest type. We provide an effective termination condition for Vinberg’s semi-algorithm with which it becomes an algorithm for finding maximal reflection sublattices. The main new ingredient of the proof is an upper bound for the number of faces of an arithmetic hyperbolic Coxeter polyhedron in terms of its volume.

Residual equidistribution of modular symbols and cohomology classes for quotients of hyperbolic 𝑛-space

We provide a new and simple automorphic method using Eisenstein series to study the equidistribution of modular symbols modulo primes, which we apply to prove an average version of a conjecture of Mazur and Rubin. More precisely, we prove that modular symbols corresponding to a Hecke basis of weight 2 cusp forms are asymptotically jointly equidistributed mod p p while we allow restrictions on the location of the cusps. As an application, we obtain a residual equidistribution result for Dedekind sums. Furthermore, we calculate the variance of the distribution and show a surprising bias with connections to perturbation theory. Additionally, we prove the full conjecture in some particular cases using a connection to Eisenstein congruences. Finally, our methods generalise to equidistribution results for cohomology classes of finite volume quotients of n n -dimensional hyperbolic space.

The Moduli Space of Points in the Boundary of Quaternionic Hyperbolic Space

Let $\mathcal{F}_1(n,m)$ be the ${\rm PSp}(n,1)$-configuration space of ordered $m$-tuple of pairwise distinct points in the boundary of quaternionic hyperbolic n-space $\partial\mathbf{H}_\mathbb{H}^n$ , i.e., the $m$-tuple of pairwise distinct points in $\partial\mathbf{H}_\mathbb{H}^n$ up to the diagonal action of ${\rm PSp}(n,1)$. In terms of Cartan's angular invariant and cross-ratio invariants, the moduli space of $\mathcal{F}_1(n,m)$ is described by using Moore's determinant. We show that the moduli space of $\mathcal{F}_1(n,m)$ is a real $2m^2-6m+5-\sum^{m-n-1}_{i=1}{m-2 \choose n-1+i}$ dimensional subset of a algebraic variety with the same real dimension when $m>n+1$.

A para-Kähler structure in the space of oriented geodesics in a real space form

In this article, we construct a new para-Kahler structure $$({{\mathcal {G}}},{{\mathcal {J}}},\Omega )$$ on the space of oriented geodesics of a n-dimensional (non-flat) real space form. We first show that the para-Kahler metric $${{\mathcal {G}}}$$ is scalar flat and when $$n=3$$ , it is locally conformally flat. Furthermore, we prove that the space of oriented geodesics of hyperbolic n-space equipped with the constructed metric $${{\mathcal {G}}}$$ is minimally isometrically embedded in the tangent bundle of hyperbolic n-space. We then study submanifold theory, and show that $${{\mathcal {G}}}$$ -geodesics correspond to minimal ruled surfaces in the real space form. Lagrangian submanifolds (with respect to the symplectic structure $$\Omega $$ ) play an important role in the geometry of the space of oriented geodesics as they come from the Gauss map of hypersurfaces in the corresponding space form. We demonstrate that the Gauss map of a non-flat hypersurface of constant Gauss curvature is a minimal Lagrangian submanifold. Finally, we show that a Hamiltonian minimal submanifold is locally the Gauss map of a hypersurface $$\Sigma $$ , which is a critical point of the functional $$\mathcal {F}(\Sigma )=\int _{\Sigma }\sqrt{|K|}\,dV$$ , K denoting the Gaussian curvature of $$\Sigma $$ .

Universal chain structure of quadratic algebras for superintegrable systems with coalgebra symmetry

In this paper we show that the chain structure of (overlapping) quadratic algebras, recently introduced in Liao et al (2018 J. Phys. A: Math. Theor. 51 255201) in the analysis of the nD quantum quasi-generalized Kepler–Coulomb system, naturally arises for nD Hamiltonian systems endowed with an coalgebra symmetry. As a consequence of this hidden symmetry, in fact, such systems are automatically endowed with 2n − 3 (second-order) functionally/algebraically independent classical/quantum conserved quantities arising as the image, through a given symplectic/differential representation, of the so-called left and right Casimirs of the coalgebra. These integrals, which are said to be being in common to the entire coalgebraic family of Hamiltonians, are shown to be the building blocks of the overlapping quadratic algebras mentioned above. For this reason a subalgebra of these quadratic structures turns out to be, as a matter of fact, universal in the sense that it is shared by any Hamiltonian belonging to this class. As a new specific result arising from this observation, we present the chain structure of quadratic algebras for the nD quasi-generalized Kepler–Coulomb system on the n-sphere and on the hyperbolic n-space . Both the classical and the quantum frameworks will be considered.

Projection theorems in hyperbolic space

We establish Marstrand-type projection theorems for orthogonal projections along geodesics onto m-dimensional subspaces of the hyperbolic n-space by a geometric argument. Moreover, we obtain a Besicovitch–Federer type characterization of purely unrectifiable sets in terms of these hyperbolic orthogonal projections.

Finite time blowup of multidimensional inhomogeneous isotropic Landau–Lifshitz equation on a hyperbolic space

Blowup solutions for the multidimensional isotropic inhomogeneous Landau-Lifshitz (ILL) equation on a hyperbolic n-space H2 are obtained. If the inhomogeneous terms are carefully selected, the ILL will guarantee three types of singularity solutions. Type I and type II blowup solutions of any dimensional ILL on H2 can develop a finite time singularity from smooth initial data with finite L2 energy. We prove the decay rate of energy density of the type III implicit blowup solution. If ε and C are non-zero constants, this solution develops a singularity in finite time with a local (r∈[ε,C]) finite initial energy.

On deformation spaces of nonuniform hyperbolic lattices

AbstractLet Γ be a nonuniform lattice acting on the real hyperbolic n-space. We show that in dimension greater than or equal to 4, the volume of a representation is constant on each connected component of the representation variety of Γ in SO(n, 1). Furthermore, in dimensions 2 and 3, there is a semialgebraic subset of the representation variety such that the volume of a representation is constant on connected components of the semialgebraic subset. Combining our approach with the main result of [2] gives a new proof of the local rigidity theorem for nonuniform hyperbolic lattices and the analogue of Soma's theorem, which shows that the number of orientable hyperbolic manifolds dominated by a closed, connected, orientable 3-manifold is finite, for noncompact 3-manifolds.

Directions in hyperbolic lattices

Abstract It is well known that the orbit of a lattice in hyperbolic n-space is uniformly distributed when projected radially onto the unit sphere. In the present work, we consider the fine-scale statistics of the projected lattice points, and express the limit distributions in terms of random hyperbolic lattices. This provides in particular a new perspective on recent results by Boca, Popa, and Zaharescu on 2-point correlations for the modular group, and by Kelmer and Kontorovich for general lattices in dimension n = 2.

The Number of Cusps of Right-angled Polyhedra in Hyperbolic Spaces

As was pointed out in (1) and (2), a right-angled polyhedra of finite volume in hyperbolic n-space H n have at least one cusp for n � 5. We obtain non-trivial lower bounds on the number of cusps. For example, right-angled polyhedra of finite volume must have at least three cusps for n = 6. Our theorem also says that the higher the dimension of right-angled polyhedron becomes, the more cusps it must have. Mathematics Subject Classification (2010). 20F55, 51F15, 57M50.

Congruence classes of points in quaternionic hyperbolic space

An important problem in quaternionic hyperbolic geometry is to classify ordered $m$-tuples of pairwise distinct points in the closure of quaternionic hyperbolic n-space, $\overline{{\bf H}_\bh^n}$, up to congruence in the holomorphic isometry group ${\rm PSp}(n,1)$ of ${\bf H}_\bh^n$. In this paper we concentrate on two cases: $m=3$ in $\overline{{\bf H}_\bh^n}$ and $m=4$ on $\partial{\bf H}_\bh^n$ for $n\geq 2$. New geometric invariants and several distance formulas in quaternionic hyperbolic geometry are introduced and studied for this problem. The congruence classes are completely described by quaternionic Cartan's angular invariants and the distances between some geometric objects for the first case. The moduli space is constructed for the second case.

The lower dimensional Busemann–Petty problem in the complex hyperbolic space

The lower dimensional Busemann–Petty problem asks whether origin-symmetric convex bodies in Rn with smaller volume of all k-dimensional sections necessarily have smaller volume. The answer is negative for k>3. The problem is still open for k=2,3. We study this problem in the complex hyperbolic n-space HCn and prove that the answer is affirmative only for sections of complex dimension one and negative for sections of higher dimensions.

Invariant distributions and cohomology for geodesic flows and higher cohomology of higher-rank Anosov actions

We are motivated by a conjecture of A. and S. Katok to study the smooth cohomologies of a family of Weyl chamber flows. The conjecture is a natural generalization of the Livšic Theorem to Anosov actions by higher-rank abelian groups; it involves a description of top-degree cohomology and a vanishing statement for lower degrees. Our main result, proved in Part II, verifies the conjecture in lower degrees for our systems, and steps in the “correct” direction in top degree. In Part I we study our “base case”: geodesic flows of finite-volume hyperbolic manifolds. We describe obstructions (invariant distributions) to solving the coboundary equation in unitary representations of the group of orientation-preserving isometries of hyperbolic N-space, and we study Sobolev regularity of solutions. (One byproduct is a smooth Livšic Theorem for geodesic flows of hyperbolic manifolds with cusps.) Part I provides the tools needed in Part II for the main theorem.

The Busemann–Petty problem in the complex hyperbolic space

AbstractThe Busemann–Petty problem asks whether origin-symmetric convex bodies in ℝn with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if n ≤ 4 and negative if n ≥ 5. We study this problem in the complex hyperbolic n-space ℍnℂ and prove that the answer is affirmative for n ≤ 2 and negative for n ≥ 3.

Hyperbolic Coxeter Pyramids

Hyperbolic Coxeter polytopes are defined precisely by combinatorial type. Polytopes in hyperbolic n-space with n + p faces that have the combinatorial type of a pyramid over a product of simplices were classified by Tumarkin for small p. In this article we generalise Tumarkin’s methods and find the remaining hyperbolic Coxeter pyramids.

Complex Boosts: A Hermitian Clifford Algebra Approach

The aim of this paper is to study complex boosts in complex Minkowski space-time that preserves the Hermitian norm. Starting from the spin group Spin $${^+(2n, 2m, \mathbb{R})}$$ in the real Minkowski space $${\mathbb{R}^{2n,2m}}$$ we construct a Clifford realization of the pseudo-unitary group U(n,m) using the space-time Witt basis in the framework of Hermitian Clifford algebra. Restricting to the case of one complex time direction we derive a general formula for a complex boost in an arbitrary complex direction and its KAK-decomposition, generalizing the well-known formula of a real boost in an arbitrary real direction. In the end we derive the complex Einstein velocity addition law for complex relativistic velocities, by the projective model of hyperbolic n-space.

Horoball packings to the totally asymptotic regular simplex in the hyperbolic n-space

In Szirmai (Acta Mathematica Hungarica 136/1-2:39–55, 2012) we generalized the notion of simplicial density function for horoballs in the extended hyperbolic space \({\overline{\mathbf{H}}^3}\), where we allowed horoballs in different types centered at various vertices of a totally asymptotic tetrahedron. By this new aspect, in this paper we study the locally densest horoball packing arrangements and their densities with respect to totally asymptotic regular simplices in the hyperbolic n-space \({\overline{\mathbf{H}}^n}\) extended with its absolute figure, where the ideal centers of horoballs lie in the vertices of a totally asymptotic regular tetrahedron. We will prove that, in this sense, the well known Boroczky–Florian density upper bound for “congruent ball and horoball” packings of\({\overline{\mathbf{H}}^3}\)does not remain valid for the analogous packing of\({\overline{\mathbf{H}}^n}\), for n ≥ 4. Although locally optimal ball arrangements do not seem to have extensions to the whole n-dimensional hyperbolic space. Moreover, we determine an explicit formula for the density of the above locally optimal horoball packings, allowing horoballs of different types (i.e. they are differently packed in their ideal simplex).

Local rigidity of hyperbolic manifolds with geodesic boundary

Let W be a compact hyperbolic n-manifold with totally geodesic boundary. We prove that if n>3 then the holonomy representation of pi_1 (W) into the isometry group of hyperbolic n-space is infinitesimally rigid.