Maximal Fuchsian groups

Maximal Fuchsian Groups

1. DEFINITIONS. Let D be the unit disk {z\ \z\ < l } and let £ be the group of conformai homeomorphisms of D. A Fuchsian group is a discrete subgroup of <£. We shall be concerned here with the finitely generated Fuchsian groups. It is known that these have the following presentations. Generators: aïy bi, • • • , ag, bg, eh • • • , ek, fa, • • • , hm, pu • * * , pr. Defining relations : e? = eg ...

Fuchsian Groups: Intro

In various ways, mathematicians associate extra structure with sets and then investigate these objects. For example, by adding a binary operation to a set and validating that the set with the binary operation fulfills certain axioms, we arrive at a group. Similarly, given a set, one can specify a collection of subsets fulfilling some criteria that gives a topology, a sense of which points are c...

Poincaré’s Theorem for Fuchsian Groups

We present a proof of Poincaré’s Theorem on the existence of a Fuchsian group for any signature (g; m1, ..., mr), where g, m1, ..., mr provide an admissable solution to the formula describing the hyperbolic area of the group’s quotient space. Along the way we elucidate relevant concepts in hyperbolic geometry and the theory of Fuchsian groups.

Compact Deformations of Fuchsian Groups

Traces in Arithmetic Fuchsian Groups