Related papers: Coxeter Decompositions of Bounded Hyperbolic Pyram…

We compute the differential geometric invariants of cuspidal edges on flat surfaces in hyperbolic $3$-space and in de Sitter space. Several dualities of invariants are pointed out.

In this survey we discuss how geometric methods can be used to study topological properties of 3-manifolds such as their Heegaard genus or the rank of their fundamental group. On the other hand, we also discuss briefly some results relating…

Generalizing both hyperbolic framed surfaces and one-parameter families of hyperbolic framed curves, we introduce the concept of hyperbolic generalized framed surfaces and establish their relations in hyperbolic 3-space. We provide the…

We provide a new formulation and proof of the triangle altitudes theorem in hyperbolic plane geometry, together with an easily computed discriminant to distinguish between different basic configurations of the altitudes of such a triangle.

We extend the concept of renormalized volume for geometrically finite hyperbolic $3$-manifolds, and show that is continuous for geometrically convergent sequences of hyperbolic structures over an acylindrical 3-manifold $M$ with…

We present explicit geometric decompositions of the complement of tiling links, which are alternating links whose projection graphs are uniform tilings of the 2-sphere, the Euclidean plane or the hyperbolic plane. This requires generalizing…

The deformation space of real projective structures parametrizes the space of the convex real projective structures on an orbifold. The Coxeter orbifold can be obtained $D^2(;n_1,n_2,n_3,n_4)\times\mathbb{R}$ by embedding the Coxeter…

In this paper, we study the problem of hyperball (hypersphere) packings in $n$-dimensional hyperbolic space ($n \ge 4$). We prove that to each $n$-dimensional congruent saturated hyperball packing, there is an algorithm to obtain a…

This paper provides an iterative procedure for constructing hyperbolic Coxeter groups that virtually fiber over $\mathbb{Z}$ that is flexible enough to yield infinitely many isomorphism classes in each virtual cohomological dimension (vcd)…

In this paper we study existence and lack thereof of closed embedded orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable hyperbolic manifolds.

We describe the deformation space of a solid torus with boundary modelled on convex ideal hyperbolic polyhedra. This deformation space is given by natural Gauss--Bonnet type inequalities on the dihedral angles. The result extends to solid…

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…

In this paper, we prove the Bounded Height Conjecture which the author formulated in [2]. As a corollary, it follows that there are only a finite number of hyperbolic three manifolds of bounded volume and trace field degree.

We show that all hyperbolic surfaces admit an ideal triangulation with bounded shear parameters. This upper bound depends logarithmically on the topology of the surface.

A knotted surface in the 4-sphere may be described by means of a hyperbolic diagram that captures the 0-section of a special Morse function, called a hyperbolic decomposition. We show that every hyperbolic decomposition of a knotted surface…

We identify and study a class of hyperbolic 3-manifolds (which we call Macfarlane manifolds) whose quaternion algebras admit a geometric interpretation analogous to Hamilton's classical model for Euclidean rotations. We characterize these…

We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of four-manifolds and extends the classical concept of bridge splittings of links in the…

This paper gives an exposition of the authors' harmonic deformation theory for 3-dimensional hyperbolic cone-manifolds. We discuss topological applications to hyperbolic Dehn surgery as well as recent applications to Kleinian group theory.…

Let $(M, \dr M)$ be a 3-manifold with incompressible boundary that admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$ such that $\dr M$ looks locally like a hyperideal polyhedron, and we characterize the…

In this paper we study discreteness of complex hyperbolic triangle groups of type $[m,m,0;3,3,2]$, i.e. groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders $3,3,2$ in complex geodesics with…