The transforms of Fuchsian groups

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...

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 ...

Compact Deformations of Fuchsian Groups

Alternating Quotients of Fuchsian Groups

It all started with a theorem of Miller [14]: the classical modular group PSL2Z‘ has among its homomorphic images every alternating group, except A6; A7; and A8. In the late 1960s Graham Higman conjectured that any (finitely generated non-elementary) Fuchsian group has among its homomorphic images all but finitely many of the alternating groups. This reduces to an investigation of the cocompac...

Genera of Arithmetic Fuchsian Groups

Introduction. The fundamental invariant of a Riemann surface is its genus. In this paper, using arithmetical means, we calculate the genus of certain Riemann surfaces defined by unit groups in quaternion algebras. First we recall a well-known general construction of Riemann surfaces. The group SL2(R) acts on the upper half-plane H by Möbius transformations. If G is a Fuchsian group, that is, a ...