Analysis of Hyperbolic Transition Curve Geometry

Hyperbolic 3-manifolds and the geometry of the curve complex

We give a brief survey of some recent work on 3-manifolds, notably towards proving Thurston’s ending lamination conjecture. We describe some applications to the theory of surfaces and mapping class groups.

Hyperbolic Geometry

Hyperbolic Geometry: Isometry Groups of Hyperbolic Space

The goal of this paper is twofold. First, it consists of an introduction to the basic features of hyperbolic geometry, and the geometry of an important class of functions of the hyperbolic plane, isometries. Second, it identifies a group structure in the set of isometries, specifically those that preserve orientation, and deals with the topological properties of their discrete subgroups. In the...

On the Generating Motions and the Convexity of a Well-Known Curve in Hyperbolic Geometry

In Euclidean geometry the vertices P of those angles ∠APB of size α that pass through the endpoints A, B of a given segment trace the arc of a circle. In hyperbolic geometry on the other hand a set of equivalently defined points P determines a different kind of curve. In this paper the most basic property of the curve, its convexity, is established. No straight-forward proof could be found. The...

Emergent Hyperbolic Network Geometry

A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a ...