Some Remarks on Non-discrete M Obius Groups

This paper contains some tentative steps towards describing the structure of non-discrete subgroups of SL(2; R). The main idea is that if a one-parameter family of groups G z varies analytically with the parameter z , then, using analytic continuation, certain results about discrete groups can be analytically continued to those groups in the family that are not discrete. The paper concentrates on families of groups generated by two parabolic transformations and, as an illustration, contains a proof that, for all but a countable set of exceptional values of the parameter z , the hyperbolic area of a hyperbolic quadilateral whose sides are paired by some pair of parabolic generators of G z is independent of the choice of generators. This is the analogue of the familiar result that the area of a fundamental region of a discrete group is independent of the choice of the generators, but it applies here to almost all non-discrete groups in the family. It is also shown that exceptional groups exist, and explicit examples of these are given. The paper ends with some unanswered questions.