Complexity and Discreteness Criteria

Let G be a subgroup of PSL(2, C). The discreteness problem for G is the problem of determining whether or not G is discrete. In this paper we assume that all groups considered are non-elementary. If G is generated by two elements, A and B of PSL(2, C), we have what is called the two-generator discreteness problem. Two-generator groups are important because, by a result of Jorgensen, an arbitrary subgroup of PSL(2, C) is discrete if and only if every non-elementary two-generator subgroup is [8]. One solution to the two-generator real discreteness problem (i.e., A and B in PSL(2, R)) is a geometrically motivated algorithm which was begun in [7] and completed in [6], where the algorithm is given in three forms. Our goal here is to compute the computational complexity of the three forms of the PSL(2, R) discreteness algorithm that appear in [6] and of the algorithm restricted to PSL(2, Q). While this may seem far afield from the original mathematical problem of determining discreteness, it settles the question as to in what sense the discreteness problem requires an algorithm. That is, the computational complexity of the algorithm can be used to prove that an algorithmic approach is necessary and that the geometric algorithm of [6] is the best discreteness condition that one can hope to obtain. We also investigate the computational complexity of several other discreteness criteria, including Riley's PSL(2, C) procedure [15], which is not always an algorithm, and Jorgensen's inequality [8]. We find that the geometric two-generator PSL(2, Q) algorithm is of linear complexity. By contrast, Riley's procedure appears to be at least exponential even when restricted to the two-generator PSL(2, Q) case; but it has never been completely analyzed.