OPTi's Algorithm for Discreteness Determination

Abstract

We summarize how OPTi draws the parameter space. Each point in the picture of a parameter space corresponds to a group, and the program colors the point according to whether the group is discrete or indiscrete. Applying Jørgensen's inequality to certain sets of generators, OPTi first tries to decide indiscreteness of the group. If the process fails for generators up to a certain depth, the program then tries to construct the Ford region. When it succeeds in constructing the Ford region, Poincaré's polyhedron theorem guarantees the discreteness of the group.