Hyperbolic Geometry Research Articles

Absolute profinite rigidity and hyperbolic geometry

We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[\omega])$ with $\omega^2+\omega+1 = 0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in $\mathrm{PSL}(2,\mathbb{C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic 3-manifold of minimal volume).

High Dimensional Hyperbolic Geometry of Complex Networks

High dimensional embeddings of graph data into hyperbolic space have recently been shown to have great value in encoding hierarchical structures, especially in the area of natural language processing, named entity recognition, and machine generation of ontologies. Given the striking success of these approaches, we extend the famous hyperbolic geometric random graph models of Krioukov et al. to arbitrary dimension, providing a detailed analysis of the degree distribution behavior of the model in an expanded portion of the parameter space, considering several regimes which have yet to be considered. Our analysis includes a study of the asymptotic correlations of degree in the network, revealing a non-trivial dependence on the dimension and power law exponent. These results pave the way to using hyperbolic geometric random graph models in high dimensional contexts, which may provide a new window into the internal states of network nodes, manifested only by their external interconnectivity.

Link prediction with hyperbolic geometry

Link prediction is a paradigmatic problem in network science with a variety of applications. In latent space network models this problem boils down to ranking pairs of nodes in the order of increasing latent distances between them. The network model with hyperbolic latent spaces has a number of attractive properties suggesting it must be a powerful tool to predict links, but the past work in this direction reported mixed results. Here we perform systematic investigation of the utility of latent hyperbolic geometry for link prediction in networks. We first show that some measures of link prediction accuracy are extremely sensitive with respect to inaccuracies in the inference of latent hyperbolic coordinates of nodes, so that we develop a new coordinate inference method that maximizes the accuracy of such inference. Applying this method to synthetic and real networks, we then find that while there exists a multitude of competitive methods to predict obvious easy-to-predict links, among which hyperbolic link prediction is rarely the best but often competitive, it is the best, often by far, when the task is to predict less obvious missing links that are really hard to predict. These links include missing links in incomplete networks with large fractions of missing links, missing links between nodes that do not have any common neighbors, and missing links between dissimilar nodes at large latent distances. Overall these results suggest that the harder a specific link prediction task is, the more seriously one should consider using hyperbolic geometry.

Central charges of aperiodic holographic tensor-network models

Central to the AdS/CFT correspondence is a precise relationship between the curvature of an anti-de Sitter (AdS) spacetime and the central charge of the dual conformal field theory (CFT) on its boundary. Our work shows that such a relationship can also be established for tensor network models of AdS/CFT based on regular bulk geometries, leading to an analytical form of the maximal central charges exhibited by the boundary states. We identify a class of tensors based on Majorana dimer states that saturate these bounds in the large curvature limit, while also realizing perfect and block-perfect holographic quantum error correcting codes. Furthermore, the renormalization group description of the resulting model is shown to be analogous to the strong disorder renormalization group, thus giving the first example of an exact quantum error correcting code that gives rise to a well-understood critical system. These systems exhibit a large range of fractional central charges, tunable by the choice of bulk tiling. Our approach thus provides a precise physical interpretation of tensor network models on regular hyperbolic geometries and establishes quantitative connections to a wide range of existing models.

Isomorphism of Binary Operations in Differential Geometry

We consider smooth binary operations invariant with respect to unitary transformations that generalize the operations of the Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry, known as Einstein addition and Möbius addition. It is shown that all such operations may be recovered from associated metric tensors that have a canonical form. Necessary and sufficient conditions for canonical metric tensors to generate binary operations are found. A definition of algebraic isomorphism of binary operations is given. Necessary and sufficient conditions for binary operations to be isomorphic are provided. It is proved that every algebraic automorphism gives rise to isomorphism of corresponding gyrogroups. Necessary and sufficient conditions in terms of metric tensors for binary operations to be isomorphic to Euclidean addition are given. The problem of binary operations to be isomorphic to Einstein addition is also solved in terms of necessary and sufficient conditions. We also obtain necessary and sufficient conditions for binary operations having the same function-parameter in the canonical representation of metric tensors to be isomorphic.

Robust Formal Watermarking Model Based on the Hyperbolic Geometry for Image Security

The digital revolution has led to an increase in the production and exchange of valuable digitized documents across institutions, companies and the general public alike. Ensuring the authenticity, integrity and ownership of these official or high-value documents is essential if they are to be considered useful. Digital watermarking is a possible solution to this challenge as it has already been used for copyright protection, source tracking, and video authentication to name just a few applications of its use. It also enables integrity protection, which is of value for numerous documents types (e.g., official documents, medical images). In this paper, we propose a new watermarking solution that is applicable to image watermarking and is based on hyperbolic geometry. Our new solution builds upon existing work in geometrical watermarking.

Large-scale influence of defect bonds in geometrically constrained self-assembly.

Recently, the importance of higher-order interactions in the physics of quantum systems and nanoparticle assemblies has prompted the exploration of new classes of networks that grow through geometrically constrained simplex aggregation. Based on the model of chemically tunable self-assembly of simplexes [Šuvakov et al., Sci. Rep. 8, 1987 (2018)2045-232210.1038/s41598-018-20398-x], here we extend the model to allow the presence of a defect edge per simplex. Using a wide distribution of simplex sizes (from edges, triangles, tetrahedrons, etc., up to 10-cliques) and various chemical affinity parameters, we investigate the magnitude of the impact of defects on the self-assembly process and the emerging higher-order networks. Their essential characteristics are treelike patterns of defect bonds, hyperbolic geometry, and simplicial complexes, which are described using the algebraic topology method. Furthermore, we demonstrate how the presence of patterned defects can be used to alter the structure of the assembly after the growth process is complete. In the assemblies grown under different chemical affinities, we consider the removal of defect bonds and analyze the progressive changes in the hierarchical architecture of simplicial complexes and the hyperbolicity parameters of the underlying graphs. Within the framework of cooperative self-assembly of nanonetworks, these results shed light on the use of defects in the design of complex materials. They also provide a different perspective on the understanding of extended connectivity beyond pairwise interactions in many complex systems.

Future Attractors in 2+1 Dimensional Λ Gravity

Any 2+1 dimensional Einsteinian spacetime with positive cosmological constant and compact hyperbolic spatial topology evolves in the expanding direction to a de Sitter model with homogeneous hyperbolic spatial geometry. In particular, the latter constitutes the uniform future attractor for this class of spacetimes for arbitrary large initial data. This is an example of large-data homogenization and demonstrates the ability of gravitation to cause late time homogenization in the evolution of cosmological models independently of their initial state.

Differential Geometry and Binary Operations

We derive a large set of binary operations that are algebraically isomorphic to the binary operation of the Beltrami–Klein ball model of hyperbolic geometry, known as the Einstein addition. We prove that each of these operations gives rise to a gyrocommutative gyrogroup isomorphic to Einstein gyrogroup, and satisfies a number of nice properties of the Einstein addition. We also prove that a set of cogyrolines for the Einstein addition is the same as a set of gyrolines of another binary operation. This operation is found directly and it turns out to be commutative. The same results are obtained for the binary operation of the Beltrami–Poincare disk model, known as Möbius addition. We find a canonical representation of metric tensors of binary operations isomorphic to the Einstein addition, and a canonical representation of metric tensors defined by cogyrolines of these operations. Finally, we derive a formula for the Gaussian curvature of spaces with canonical metric tensors. We obtain necessary and sufficient conditions for the Gaussian curvature to be equal to zero.

The Siegel-Klein Disk: Hilbert Geometry of the Siegel Disk Domain.

We study the Hilbert geometry induced by the Siegel disk domain, an open-bounded convex set of complex square matrices of operator norm strictly less than one. This Hilbert geometry yields a generalization of the Klein disk model of hyperbolic geometry, henceforth called the Siegel–Klein disk model to differentiate it from the classical Siegel upper plane and disk domains. In the Siegel–Klein disk, geodesics are by construction always unique and Euclidean straight, allowing one to design efficient geometric algorithms and data structures from computational geometry. For example, we show how to approximate the smallest enclosing ball of a set of complex square matrices in the Siegel disk domains: We compare two generalizations of the iterative core-set algorithm of Badoiu and Clarkson (BC) in the Siegel–Poincaré disk and in the Siegel–Klein disk: We demonstrate that geometric computing in the Siegel–Klein disk allows one (i) to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in the Siegel–Poincaré disk model, and (ii) to approximate fast and numerically the Siegel–Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries.

Holography on tessellations of hyperbolic space

We compute boundary correlation functions for scalar fields on tessellations of two- and three-dimensional hyperbolic geometries. We present evidence that the continuum relation between the scalar bulk mass and the scaling dimension associated with boundary-to-boundary correlation functions survives the truncation of approximating the continuum hyperbolic space with a lattice.

COMBINATORIAL INSCRIBABILITY OBSTRUCTIONS FOR HIGHER DIMENSIONAL POLYTOPES

For $3$-dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hyperbolic geometry. In particular, inscribability can be tested in polynomial time, and for every $f$-vector of $3$-polytopes, there exists an inscribable polytope with that $f$-vector. For higher-dimensional polytopes, much less is known. Of course, for any inscribable polytope, all of its lower-dimensional faces need to be inscribable, but this condition does not appear to be very strong. We observe non-trivial new obstructions to the inscribability of polytopes that arise when imposing that a certain inscribable face be inscribed. Using this obstruction, we show that the duals of the $4$-dimensional cyclic polytopes with at least $8$ vertices---all of whose faces are inscribable---are not inscribable. This result is optimal in the following sense: We prove that the duals of the cyclic $4$-polytopes with up to $7$ vertices are, in fact, inscribable. Moreover, we interpret this obstruction combinatorially as a forbidden subposet of the face lattice of a polytope, show that $d$-dimensional cyclic polytopes with at least $d+4$ vertices are not circumscribable, and that no dual of a neighborly $4$-polytope with $8$ vertices, that is, no polytope with $f$-vector $(20,40,28,8)$, is inscribable.

Comments on the Kramer Kramer Man’ko 2 × 2 matrix multiplication

This article is a complement to a paper by Kramer et al. that proposes a new matrix product, which is like a mixture of the usual and the Hadamard (or Schur) product. This new product is then related to the transformation group in the hyperbolic plane. Actually variations in the definition of internal operations lead to an infinity of products that can be associated with any of the Euclidean, hyperbolic or elliptic geometries in quite different manners. By the way: This article is not a review. The review can be found at the end of the paper.

Topological Hyperbolic Lattices.

Non-Euclidean geometry, discovered by negating Euclid's parallel postulate, has been of considerable interest in mathematics and related fields for the description of geographical coordinates, Internet infrastructures, and the general theory of relativity. Notably, an infinite number of regular tessellations in hyperbolic geometry-hyperbolic lattices-are expected to extend Euclidean Bravais lattices and the consequent wave phenomena to non-Euclidean geometry. However, topological states of matter in hyperbolic lattices have yet to be reported. Here we investigate topological phenomena in hyperbolic geometry, exploring how the quantized curvature and edge dominance of the geometry affect topological phases. We report a recipe for the construction of a Euclidean photonic platform that inherits the topological band properties of a hyperbolic lattice under a uniform, pseudospin-dependent magnetic field, realizing a non-Euclidean analog of the quantum spin Hall effect. For hyperbolic lattices with different quantized curvatures, we examine the topological protection of helical edge states and generalize Hofstadter's butterfly, by employing two empirical parameters that measure the edge confinement and defect immunity. We demonstrate that the proposed platforms exhibit the unique spectral-magnetic sensitivity of topological immunity in highly curved hyperbolic planes. Our approach is applicable to general non-Euclidean geometry and enables the exploitation of infinite lattice degrees of freedom for band theory.

On freedman equation and the shape of our universe

In the nineteen twenties, the famous Russian mathematician Alexander Freedman formulated an equation which determines the shape and fate of our universe. Freedman derived his equation in general relativity. The equation reveals that the geometry of the universe may be flat, closed or open. The Euclidean, hyperbolic and spherical geometries describe the flat, open and closed universes respectively. Both NASA’s WMAP and ESA’s PLANCK mission show the cosmological curvature parameter, ΩK, to be 0.000±0.005, consistent with a flat universe. Many observational cosmological probes revealed that the universe is flat obeying the classical Euclidean geometry. But till this day, there is no mathematical formulation/proof for the geometry of our universe. In this work, the attempts to establish that the shape of our universe is flat.

Binary Operations in the Unit Ball: A Differential Geometry Approach

Within the framework of differential geometry, we study binary operations in the open, unit ball of the Euclidean n-space R n , n ∈ N , and discover the properties that qualify these operations to the title addition despite the fact that, in general, these binary operations are neither commutative nor associative. The binary operation of the Beltrami-Klein ball model of hyperbolic geometry, known as Einstein addition, and the binary operation of the Beltrami-Poincaré ball model of hyperbolic geometry, known as Möbius addition, determine corresponding metric tensors in the unit ball. For a variety of metric tensors, including these two, we show how binary operations can be recovered from metric tensors. We define corresponding scalar multiplications, which give rise to gyrovector spaces, and to norms in these spaces. We introduce a large set of binary operations that are algebraically equivalent to Einstein addition and satisfy a number of nice properties of this addition. For such operations we define sets of gyrolines and co-gyrolines. The sets of co-gyrolines are sets of geodesics of Riemannian manifolds with zero Gaussian curvatures. We also obtain a special binary operation in the ball, which is isomorphic to the Euclidean addition in the Euclidean n-space.

Hyperbolic Superspaces and Super-Riemann Surfaces

In this paper, we will generalize some results in Manin's paper "Three-dimensional hyperbolic geometry as $\infty$-adic Arakelov geometry" to the supergeometric setting. More precisely, viewing $\mathbb{C}^{1|1}$ as the boundary of the hyperbolic superspace $\mathcal{H}^{3|2}$, we reexpress the super-Green functions on the supersphere $\hat{\mathbb{C}}^{1|1}$ and the supertorus $T^{1|1}$ by some data derived from the supergeodesics in $\mathcal{H}^{3|2}$.

Hyperbolic flowers

ABSTRACT Crochet models of a hyperbolic plane are very popular as they help to visualize complicated objects in hyperbolic geometry. We present another way to make crochet models as a part of a triangulated plane and also provide an example of a cylinder in hyperbolic space. This approach assists in understanding of various properties of hyperbolic geometry such as the relation between edges and angles in a hyperbolic triangle, tiling of a hyperbolic plane or Nash–Kuiper embedding theorem.

Polygons in hyperbolic geometry using Beltrami-Klein models

Hyperbolic geometry was a geometry based on Hyperbolic Parallel Postulate. The purpose of this research was to describe the model of basic objects, concepts of triangles and polygons, as well explain types of polygons in hyperbolic geometry used the Beltrami-Klein model. The research method was literature study, it studied the definition and axiom were related to Euclid geometry and hyperbolic geometry such as point, line, triangle, and polygon. The result of this study was: 1) The plane model for Beltrami-Klein model is a γ circle. The object of geometry such as Klein points was expressed as a dot, Klein lines was expressed as part of Euclid lines that contained in a γ circle, Klein distances and Klein angles has the same presentation as Euclid distances and Euclid angles. 2) The Klein triangles have different concepts from the Euclid triangles. 3) The Klein polygons were defined as a combination of Klein segments, Klein rays, or Klein lines that was foreign to each other and have n different Klein points that was not collinearities. 4) The Klein polygons were divided into two types, the ordinary Klein polygons and the asymptotic Klein polygons.

Poincar\xe9 maps for analyzing complex hierarchies in single-cell data

The need to understand cell developmental processes spawned a plethora of computational methods for discovering hierarchies from scRNAseq data. However, existing techniques are based on Euclidean geometry, a suboptimal choice for modeling complex cell trajectories with multiple branches. To overcome this fundamental representation issue we propose Poincaré maps, a method that harness the power of hyperbolic geometry into the realm of single-cell data analysis. Often understood as a continuous extension of trees, hyperbolic geometry enables the embedding of complex hierarchical data in only two dimensions while preserving the pairwise distances between points in the hierarchy. This enables the use of our embeddings in a wide variety of downstream data analysis tasks, such as visualization, clustering, lineage detection and pseudotime inference. When compared to existing methods — unable to address all these important tasks using a single embedding — Poincaré maps produce state-of-the-art two-dimensional representations of cell trajectories on multiple scRNAseq datasets.