Modular circle quotients and PL limit sets

We say that a collection Gamma of geodesics in the hyperbolic plane H^2 is a modular pattern if Gamma is invariant under the modular group PSL_2(Z), if there are only finitely many PSL_2(Z)-equivalence classes of geodesics in Gamma, and if each geodesic in Gamma is stabilized by an infinite order subgroup of PSL_2(Z). For instance, any finite union of closed geodesics on the modular orbifold H^2/PSL_2(Z) lifts to a modular pattern. Let S^1 be the ideal boundary of H^2. Given two points p,q in S^1 we write pq if p and q are the endpoints of a geodesic in Gamma. (In particular pp.) We show that is an equivalence relation. We let Q_Gamma=S^1/ be the quotient space. We call Q_Gamma a modular circle quotient. In this paper we will give a sense of what modular circle quotients `look like' by realizing them as limit sets of piecewise-linear group actions