Computing with matrix groups over infinite rings is much more complicated then working with matrix groups over finite fields. For example, the membership problem is undecidable in general.

However, for subgroups of PSL(2, R) the situation is much betterr since we have geometric methods at hand.

In this talk, we will recall Rosenberger's classification of the discrete free subgroups G = < A, B> of PSL(2, R). Further, we will discuss how to solve the membership problem for such a group G constructively. This is joint work with B. Eick and C. Leedham-Green.