Counting and Equidistribution of Reciprocal Geodesics and Dihedral Groups

Abstract

Abstract We study the growth of the number of conjugacy classes of infinite dihedral subgroups of lattices in $\operatorname{PSL}_{2}{\mathbb{R}}$, generalizing the earlier work of Sarnak [ 9] and Bourgain–Kontorovich [ 4] on the growth of the number of reciprocal geodesics on the modular surface. We also prove that reciprocal geodesics are equidistributed in the unit tangent bundle.