The action of the modular group SL₂([special characters omitted]) on [special characters omitted] by fractional linear transformations has exactly one orbit and encodes the classical Euclidean algorithm. Long and Reid sought discrete, nonarithmetic subgroups of SL₂([special characters omitted]) that also act transitively on [special characters omitted] through fractional linear transformations. By considering a particular two-parameter family Δ of discrete subgroups of SL₂([special characters omitted])—more specifically, Fricke groups—they succeeded in their search. They dubbed the four groups they discovered “pseudomodular”.

In this thesis we identify sufficient conditions for which a group in the family Δ has more than one orbit in its action on [special characters omitted] and is thus not pseudomodular. These conditions are that the aforementioned two parameters, which are rational numbers, either have particular congruence properties or specific number-theoretical relations.

Our main technique is as follows. The groups in Δ are Fuchsian groups whose cusps form one orbit that is a subset of [special characters omitted]. We endow [special characters omitted] with a topology that is given in a certain way by some p-adic fields and ask when the cusp sets of particular groups in Δ are dense in this topology. For each of our chosen topologies, if the set of cusps [special characters omitted] of a particular group in Δ is not dense, then [special characters omitted] cannot equal [special characters omitted], so the group in question cannot be pseudomodular. Through this method, we find several families of groups in Δ that are not pseudomodular and, along the way, we show that cusp sets admit numerous types of behavior with respect to p-adic topologies.