2-GENERATOR DISCRETE-SUBGROUPS OF PSL(2,R)

Let A and B be elements of PSL(2, R) and let G be the group they gener ate. Assume that G is non-elementary. The discreteness problem is the problem of finding an algorithm to determine whether G is or is not di screte. This is an old and subtle problem. Historically, papers on the subject have been known for their errors and omissions. In this monog raph we provide what one hopes is the definitive solution to the discr eteness problem. We present the first complete geometric solution to t he discreteness problem building upon the cases done by Gilman and Mas kit. We are able to explain exactly why the discreteness problem requi res an algorithmic solution. We translate the geometric algorithm into a a purely computational algorithm, one which allows all standard com putations with real numbers. We call this the real number algorithm. I t is a modified BSS machine. We also provide conditions on the entries of the two matrices that assure that the geometric algorithm and the real number algorithm translate into a Turing machine algorithm, an al gorithm that can be implemented on a computer.