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  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 , , , , , , , , , , , and  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 .
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 
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.