arXiv research

The study classifies all compact hyperbolic polytopes with eight facets.

Geometric constraints help classify hyperbolic polytopes.

The study classifies all compact 5D polytopes with 9 facets.

The study classifies 331 specific 4D polytopes with 7 facets.

New methods classify hyperbolic polytopes with up to 40 facets.

The paper classifies compact hyperbolic Coxeter polytopes and improves upper bounds.

New noncompact Coxeter polytopes found in various dimensions.

Researchers prove rigidity of convex polytopes in hyperbolic space using spinor techniques.

We construct infinite series of non-simple ideal hyperbolic Coxeter 4-polytopes whose growth rates are Perron numbers. This infinite series is the first example of such a non-compact infinite polytopal series.

The study of symmetries in manifolds derived from colored polytopes.

By gluing together the sides of eight copies of an all-right angled hyperbolic 6-dimensional polytope, two orientable hyperbolic 6-manifolds with Euler characteristic -1 are constructed. They are the first known examples of orientable hyperbolic 6-manifolds having the smallest possible volume.

Study of infinitesimal rigidity in hyperbolic manifolds.

We prove that among four-dimensional ideal right-angled hyperbolic polytopes the 24-cell is of minimal volume and of minimal facet number. As a corollary, a dimension bound for ideal right-angled hyperbolic polytopes is obtained.

We provide the first examples of geometric transition from hyperbolic to anti-de Sitter structures in dimension four, in a fashion similar to Danciger's three-dimensional examples. The main ingredient is a deformation of hyperbolic 4-polytopes, discovered by Kerckhoff and Storm, eventually collapsing to a 3-dimensional…

We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure. Our results are applied in spherical and hyperbolic space. This…

New hyperbolic manifolds discovered that fiber algebraically up to dimension 8.

The study constructs links from polytope subgraphs and proves their hyperbolic properties.

In this paper, we compute the covolume of the group of units of the quadratic form f_d^n(x) = x_1^2 + x_2^2 + . . . + x_n^2 - d x_{n+1}^2 with d an odd, positive, square-free integer. Mcleod has determined the hyperbolic Coxeter fundamental domain of the reflection subgroup of the group of units of the quadratic form f…

By gluing together copies of an all-right angled Coxeter polytope a number of open hyperbolic 6-manifolds with Euler characteristic -1 are constructed. They are the first known examples of hyperbolic 6-manifolds having the smallest possible volume.

Researchers prove finiteness of integral representations on specific polytopes.

This survey paper contains an elementary exposition of Casson and Rivin's technique for finding the hyperbolic metric on a 3-manifold M with toroidal boundary. We also survey a number of applications of this technique. The method involves subdividing M into ideal tetrahedra and solving a system of gluing equations to f…

Built the smallest non-commensurable hyperbolic 4-manifold.

Study of Horn's problem in PU(n,1) for n≥1.

A family of closed manifolds is called cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. We establish cohomological rigidity for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes. We consider the class P of 3…

Geodesic simplices in pseudo-hyperbolic space get a cohomological treatment.

The paper sets new limits on hyperbolic polyhedra volumes.

Study character varieties of a Coxeter group in hyperbolic and Anti-de Sitter spaces.

In this paper, for each finite group $G$, we construct explicitly a non-compact complete finite-volume arithmetic hyperbolic $4$-manifold $M$ such that $\mathrm{Isom}\,M \cong G$, or $\mathrm{Isom}^{+}\,M \cong G$. In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic $4$-space, on o…

A Coxeter $n$-orbifold is an $n$-dimensional orbifold based on a polytope with silvered boundary facets. Each pair of adjacent facets meet on a ridge of some order $m$, whose neighborhood is locally modeled on ${\mathbb R}^n$ modulo the dihedral group of order $2m$ generated by two reflections. For $n \geq 3$, we study…

The study finds all possible 3D polytopes in Riemannian 3-manifolds with positive scalar curvature.

Study on hyperbolic polyhedra and their volume, proving finiteness of arithmetic groups.

We prove that every complete finite-volume hyperbolic 3-manifold $M$ that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable hyperbolic 4-manifold $W$, which is also tessellated into right-angled regular pol…

We prove the theorem mentioned in the title, for ${\mathbb{R}}^n$, where $n \ge 3$. The case of the simplex was known previously. Also, the case $n=2$ was settled, but there the infimum was some well-defined function of the side lengths. We also consider the cases of spherical and hyperbolic $n$-spaces. There we give s…

The paper studies the topology and geometry of simple orbifolds, generalizing concepts from simple polytopes.

Asymptotic results for weighted floating bodies are established and used to obtain new proofs for the existence of floating areas on the sphere and in hyperbolic space and to establish the existence of floating areas in Hilbert geometries. Results on weighted best and random approximation and the new approach to floati…

If we fix the angles at the vertices of a convex planar $n$-gon, the lengths of its edges must satisfy two linear constraints in order for it to close up. If we also require unit perimeter, our vectors of $n$ edge lengths form a convex polytope of dimension $n-3$, each facet of which consists of those $n$-gons in which…

In this paper, we classify all the orientable hyperbolic 5-manifolds that arise as a hyperbolic space form $H^5/Γ$ where $Γ$ is a torsion-free subgroup of minimal index of the congruence two subgroup $Γ^5_2$ of the group $Γ^5$ of positive units of the Lorentzian quadratic form $x_1^2+...+x_5^2-x_6^2$. We also show that…

We develop a way of seeing a complete orientable hyperbolic $4$-manifold $\mathcal{M}$ as an orbifold cover of a Coxeter polytope $\mathcal{P} \subset \mathbb{H}^4$ that has a facet colouring. We also develop a way of finding totally geodesic sub-manifolds $\mathcal{N}$ in $\mathcal{M}$, and describing the result of mu…

Let $P$ be a (non necessarily convex) embedded polyhedron in $\R^3$, with its vertices on an ellipsoid. Suppose that the interior of $P$ can be decomposed into convex polytopes without adding any vertex. Then $P$ is infinitesimally rigid. More generally, let $P$ be a polyhedron bounding a domain which is the union of p…

New groups $Γ_n^k$ connect braids and 3-manifolds.

The study broadens the concept of cyclic polytopes to Veronese polytopes.

We characterization hyperbolic metrics on compact surfaces with boundary using a variational principle. As a consequence, a new parametrization of the Teichmuller space of compact surface with boundary is produced. In the new parametrization, the Teichmuller space becomes an open convex polytope. It is conjectured that…

The paper studies deformation spaces of Coxeter truncation polytopes.

Neural networks approximate unit spheres as polytopes.

Proves stability in Weyl polytopes using optimal transport.

Faces of quasi-arithmetic Coxeter polytopes are also quasi-arithmetic.

Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes sharing a common facet is convex. We prove that the real projective structure on…

This article announces the completion of the classification of rank 4 locally projective polytopes and their quotients. There are seventeen universal locally projective polytopes (nine nondegenerate). Amongst their 441 quotients are a further four (nonuniversal) regular polytopes, and 152 nonregular but section regular…