Counting and equidistribution of reciprocal geodesics and dihedral groups

TL;DR

This paper investigates the growth and distribution of reciprocal geodesics associated with dihedral groups in lattices of PSL(2,R), extending previous results on geodesic counts and proving their equidistribution.

Contribution

It generalizes earlier work by analyzing conjugacy classes of dihedral subgroups and establishes equidistribution of reciprocal geodesics in the unit tangent bundle.

Findings

Growth rate of conjugacy classes of dihedral subgroups quantified.

Reciprocal geodesics are proven to be equidistributed in the unit tangent bundle.

Extends previous results on geodesic growth to a broader class of groups.

Abstract

We study the growth of the number of conjugacy classes of infinite dihedral subgroups of lattices in PSL(2,R), generalizing earlier work of Sarnak and Bourgain-Kontorovich 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.