Chapter 1

Introduction

Let A and B be elements of PSL(2, R) and let G = (A, B) be the group

they generate. Assume that G is non-elementary. This

monograph1

is

devoted to the question of how to determine whether or not G is a

discrete

group2.

While J0rgensen's inequality [17] gives an elegant necessary con-

dition for discreteness in PS7y(2,C), a sufficient condition even in

PSL(2, R) has been elusive. There are numerous papers on the subject

(see [20], [22], [27], [29], [30], [31], [19], [32], [34], [35], [36], and [37] and

the rest of the bibliography), but many of these papers have errors and

omissions.

There are basically two approaches to the problem, one mostly al-

gebraic and the other involving a combination of geometry and algebra.

Much of the more algebraic treatment is due to Purzitsky and Rosen-

berger. Rosenberger gives a summary of the algebraic results in [37].

His paper cites results from more than a dozen other papers. Even

though each of the errors or omissions in the cited papers appears to

be corrected in a subsequent paper, it is difficult for a nonexpert or

even an expert to trace the complete algebraic argument. Matelski [27]

began the more geometric treatment of the problem, but his work is

incomplete.

We pursue the second approach here. The problem is divided into

Supported in part by NSF grants # DMS-9001881 & # DMS-9409115.

2Received by the editor December 1991 and in revised form March 1993, March

1994 and May 1994.

3