Equidistribution of $q$-orbits of closed geodesics

Given an element of (\mathbb{Z}/q\mathbb{Z})^{\times} we associate a closed geodesic on the modular surface. We prove that the closed geodesics associated to cosets of 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 fields as extended to subgroups by Popa. Finally, we show that the homology classes of the q -orbits of oriented closed geodesics concentrate around the Eisenstein line and present group theoretic applications.