Chapter 1
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
devoted to the question of how to determine whether or not G is a
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
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
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.
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