Related papers: The smallest Haken hyperbolic polyhedra

This article explores closed geodesics on hyperbolic surfaces. We show that, for sufficiently large $k$, the shortest closed geodesics with at least $k$ self-intersections, taken among all hyperbolic surfaces, all lie on an ideal pair of…

We consider a compact hyperbolic tetrahedron of a general type. It is a convex hull of four points called vertices in the hyperbolic space $\mathbb{H}^3$. It can be determined by the set of six edge lengths up to isometry. For further…

We describe an algorithm that, given a 3-manifold M, outputs a finite set containing all minimal volume k-component hyperbolic link complements in M. A key step, that might be of independent interest, is an algorithm that, given two…

We investigate lower bounds for the number of ideal and finite vertices of right-angled hyperbolic polyhedra of finite volume. We use a geometric method of orthogonal gluings to establish new bounds in low dimensions, specifically…

We observe that fullerene graphs are one-skeletons of polyhedra, which can be realized with all dihedral angles equal to $\pi/2$ in a hyperbolic 3-dimensional space. One of the most important invariants of such a polyhedron is its volume.…

We identify all hyperbolic knots whose complements are in the census of orientable one-cusped hyperbolic manifolds with eight ideal tetrahedra. We also compute their Jones polynomials.

The goal of this paper to determine the optimal horoball packing arrangements and their densities for all four fully asymptotic Coxeter tilings (Coxeter honeycombs) in hyperbolic 3-space $\mathbb{H}^3$. Centers of horoballs are required to…

We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space H^n for every n < 20 (resp. n < 7). When n=7 or 8, they may be taken to be nonarithmetic. Furthermore, for 1 < n <…

In this paper, we show the existence of smoothly embedded closed minimal surfaces in infinite volume hyperbolic $3$-manifolds except some special cases.

We give a cohomological interpretation of the geodesic simplices of the pseudo-hyperbolic space of signature $(p,q)$ and formulate a necessary and sufficient condition for such a simplex to have finite volume. As a corollary, we obtain that…

A connection between real poles of the growth functions for Coxeter groups acting on hyperbolic space of dimensions three and greater and algebraic integers is investigated. In particular, a geometric convergence of fundamental domains for…

In this paper, we classify all of the five-sided three-dimensional hyperbolic polyhedra with one ideal vertex, which have the shape of a triangular prism. We show how to find each such polyhedron in the upper half-space model by considering…

By work of Uhlenbeck, the largest principal curvature of any least area fiber of a hyperbolic $3$-manifold fibering over the circle is bounded below by one. We give a short argument to show that, along certain families of fibered hyperbolic…

It is well known that an arbitrary closed orientable $3$-manifold can be realized as the unique boundary of a compact orientable $4$-manifold, that is, any closed orientable $3$-manifold is cobordant to zero. In this paper, we consider the…

A convex body $R$ in the hyperbolic plane is reduced if any convex body $K\subset R$ has a smaller minimal width than $R$. We answer a few of Lassak's questions about ordinary reduced polygons regarding its perimeter, diameter and…

This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the…

On finite-volume hyperbolic $3$-manifolds, we compare volumes of different metrics using the exponential convergence of Ricci-DeTurck flow toward the hyperbolic metric $h_0$. We prove that among metrics with scalar curvature bounded below…

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 give the sharp lower bound of the volume product of three dimensional convex bodies which are invariant under a discrete subgroup of $O(3)$ in several cases. We also characterize the convex bodies with the minimal volume product in each…

A compact hyperbolic "cobweb" manifold (hyperbolic space form) of symbol $Cw(6,6,6)$ will be constructed in Fig.1,4,5 as a representant of a presumably infinite series $Cw(2p,2p,2p)$ $(3 \le p \in \bN$ natural numbers). This is a by-product…