Counting and Equidistribution of Reciprocal Geodesics and Dihedral Groups

Abstract

We study the growth of the number of conjugacy classes of infinite dihedral subgroups of lattices in PSL₂ 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.