Equidistribution of $q$-orbits of closed geodesics

We introduce a natural way of associating oriented closed geodesics on the modular curve to elements of $(\mathbb{Z}\mathrm{/}q\mathbb{Z}{)}^{\times }$ and prove that the corresponding packets associated to sufficiently large subgroups equidistribute in the unit tangent bundle as $q$ tends to infinity. This is a $q$-orbit analogue of Duke’s Theorem for real quadratic field as extended to subgroups by Popa. We also show that the homology classes of the $q$-orbits of oriented closed geodesics concentrate around the Eisenstein line and present group theoretic applications thereof.