A Note on Discrete Groups (original) (raw)

A remark on some fuchsian groups

New Trends in Mathematical Science

In this paper we study combinatorial structures of some Fuchsian groups. We examine fundamental domains, group actions and genus for these groups.

On the Geometry of Fuchsian Groups

2015

In this Master thesis we consider the discrete groups with emphasis on the geometry of discrete groups, which lie at the intersection between Hyperbolic Geometry, Topology, Abstract Algebra, and Complex Analysis. Fuchsian groups are discrete subgroups of the group PSL(2,R) of linear fractional transformations of one complex variable, which is isomorphic to a quotient topological group: PSL(2,R)≅SL(2,R)/{±I}. Here SL (2,R) is special linear group and I is the identity. We study discrete groups, in particular, Fuchsian groups. We present the geometric properties of Fuchsian groups such as fundamental domains, compactness, and Dilichlet tessellations. In addition, we also present some algebraic properties of Fuchsian groups.

A Survey On Non Characteristic Heisenberg Group Domains

arXiv: Differential Geometry, 2019

In this work, we give a survey on non characteristic domains of Heisenberg groups. We prove that bounded domains which are diffeomorphic to the solid torus having the center of the group as rotation axis, are non characteristic. Then, we state the following conjecture : The bounded non characteristic domains of the Heisenberg group of dimension 1 are those diffeomorphic to a solid torus having the center of the group as rotation axis.

KLEINIAN GROUPS WITH REAL PARAMETERS

Communications in Contemporary Mathematics, 2001

We find all real points of the analytic space of two generator Möbius groups with one generator elliptic of order two. Geometrically this is a certain slice through the space of two generator discrete groups, analogous to the Riley slice, though of a very different nature. We obtain applications concerning the general structure of the space of all two generator Kleinian groups and various universal constraints for Fuchsian groups.

A Crash Course on Kleinian Groups

Lecture Notes in Mathematics, 1974

These notes formed part of the ICTP summer school on Geometry and Topology of 3-manifolds in June 2006. Assuming only a minimal knowledge of hyperbolic geometry, the aim was to provide a rapid introduction to the modern picture of Kleinian groups. The subject has recently made dramatic progress with spectacular proofs of the Density Conjecture, the Ending Lamination Conjecture and the Tameness Conjecture. Between them, these three new theorems make possible a complete geometric classification of all hyperbolic 3-manifolds. The goal is to explain the background needed to appreciate the statements and significance of these remarkable results.

The Geometry and Arithmetic of Kleinian Groups

2013

In this article we survey and describe various aspects of the geometry and arithmetic of Kleinian groups -discrete nonelementary groups of isometries of hyperbolic 3-space. In particular we make a detailed study of two-generator groups and discuss the classification of the arithmetic generalised triangle groups (and their near relatives). This work is mainly based around my collaborations over the last two decades with Fred Gehring and Colin Maclachlan, both of whom passed away in 2012. There are many others involved as well.

Dirichlet-Ford domains and Double Dirichlet domains

Bulletin of the Belgian Mathematical Society - Simon Stevin, 2016

We continue investigations started by Lakeland on Fuchsian and Kleinian groups which have a Dirichlet fundamental domain that also is a Ford domain in the upper half-space model of hyperbolic 2-and 3-space, or which have a Dirichlet domain with multiple centers. Such domains are called DF-domains and Double Dirichlet domains respectively. Making use of earlier obtained concrete formulas for the bisectors defining the Dirichlet domain of center i ∈ H 2 or center j ∈ H 3 , we obtain a simple condition on the matrix entries of the side-pairing transformations of the fundamental domain of a Fuchsian or Kleinian group to be a DF-domain. Using the same methods, we also complement a result of Lakeland stating that a cofinite Fuchsian group has a DF domain (or a Dirichlet domain with multiple centers) if and only if it is an index 2 subgroup of the discrete group G of reflections in a hyperbolic polygon. * The first author is supported by Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Flanders), the second by FAPESP-Brazil (while visiting the Vrije Universiteit Brussel), the third by Fonds voor Wetenschappelijk Onderzoek (Flanders)-Belgium (while visiting Universität Bielefeld) and the fourth by FAPESP and CNPq-Brazil, and the fifth by FAPESP (Fundação

An Overview of Complex Kleinian Groups

Nonlinear Systems and Complexity, 2015

Classical Kleinian groups are discrete subgroups of P SL(2, C) acting on the complex projective line P 1 C , which actually coincides with the Riemann sphere, with non-empty region of discontinuity. These can also be regarded as the monodromy groups of certain differential equations. These groups have played a major role in many aspects of mathematics for decades, and also in physics. It is thus natural to study discrete subgroups of the projective group P SL(n, C), n > 2. Surprisingly, this is a branch of mathematics which is in its childhood, and in this article we give an overview of it.