Nondiscrete groups of M"obius transformations

Date of Completion

January 1996






Discrete groups of Mobius transformations may be studied using the geometry on the resulting quotients of hyperbolic space; nondiscrete groups require different methods since the quotient spaces do not carry traditional geometric structures. After a discussion of some basic concepts in the context of nondiscrete groups, we present an algorithm to find an infinite-order elliptic element in a large class of two-generator, nondiscrete, nonelementary groups of real Mobius transformations. In doing so we begin with a discreteness algorithm of Gilman and Maskit; where that algorithm stops with a result of nondiscreteness, we continue to actually pick out one of the infinitesimal rotations whose existence is guaranteed by a theorem of Jorgensen.^ We then present a computer program for visualizing limit sets of discrete groups and partial limit sets of nondiscrete groups. We discuss various experimental observations based on the output of the program, along with suggestions for future applications. ^