Topics in Hyperbolic Geometry

Metric and periodic lines in the Poincare ball model of hyperbolic geometry

In this paper, we prove that every metric line in the Poincare ball model of hyperbolic geometry is exactly a classical line of itself. We also proved nonexistence of periodic lines in the Poincare ball model of hyperbolic geometry.

An Extension of Poincare Model of Hyperbolic Geometry with Gyrovector Space Approach

‎The aim of this paper is to show the importance of analytic hyperbolic geometry introduced in [9]‎. ‎In [1]‎, ‎Ungar and Chen showed that the algebra of the group $SL(2,mathbb C)$ naturally leads to the notion of gyrogroups ‎and gyrovector spaces for dealing with the Lorentz group and its ‎underlying hyperbolic geometry‎. ‎They defined the Chen addition and then Chen model of hyperbolic geomet...

Topics in Modern geometry Second set of topics

The goal of the second set of topics in the course is to introduce students to hyperbolic geometry and its connections to dynamical systems and number theory. We build on abstract notions of topological spaces and groups that come together in this beautiful and very concrete subject. We will mostly deal with isometries of the hyperbolic plane, with projects covering connections to number theory...

Some Topics on Hyperbolic Geometry and Heegaard Splittings of 3-Manifolds

Topics in Absolute Anabelian Geometry I: Generalities

This paper forms the first part of a three-part series in which we treat various topics in absolute anabelian geometry from the point of view of developing abstract algorithms, or “software”, that may be applied to abstract profinite groups that “just happen” to arise as [quotients of] étale fundamental groups from algebraic geometry. In the present paper, after studying various abstract combin...

Topics Surrounding the Anabelian Geometry of Hyperbolic Curves

§0. Introduction §1. The Tate Conjecture as a Sort of Grothendieck Conjecture §1.1. The Tate Conjecture for non-CM Elliptic Curves §1.2. Some Pro-p Group Theory §2. Hyperbolic Curves as their own “Anabelian Albanese Varieties” §2.1. A Corollary of the Main Theorem of [Mzk2] §2.2. A Partial Generalization to Finite Characteristic §3. Discrete Real Anabelian Geometry §3.1. Real Complex Manifolds ...