Abstract: I will discuss the growth of the number of infinite dihedral subgroups of lattices G in PSL(2, R). Such subgroups exist whenever the lattice has 2-torsion and they are related to so-called reciprocal geodesics on the corresponding quotient orbifold. These are closed geodesics passing through an even order orbifold point, or equivalently, homotopy classes of closed curves having a representative in the fundamental group thatâ€™s conjugate to its own inverse. We obtain the asymptotic growth of the number of reciprocal geodesics (or infinite dihedral subgroups) in any orbifold, generalizing earlier work of Sarnak and Bourgain-Kontorivich on the growth of the number of reciprocal geodesics on the modular surface. Time allowing, I will explain how our methods also show that reciprocal geodesics are equidistributed in the unit tangent bundle. This is joint work with Juan Souto.

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Last updated: 31 May 2023