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.

arXiv comment43 pages, several minor issues have been resolved and improvements have been made. Accepted in Commentarii Mathematici Helvetici