Equidistribution of $q$-orbits of closed geodesics

Abstract

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.