The count of reciprocal conjugacy classes in Hecke groups with word length at most x is asymptotic to a dominant geometric sum plus a normal CDF correction, with explicit parameters for each p.
Counting and equidistribution of strongly reversible closed geodesics in negative curvature
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abstract
Let $M$ be a pinched negatively curved Riemannian orbifold, whose fundamental group has torsion of order $2$. Generalizing results of Sarnak and Erlandsson-Souto for constant curvature oriented surfaces, and with very different techniques, we give an asymptotic counting result on the number of strongly reversible periodic orbits of the geodesic flow in $M$, and prove their equidistribution towards the Bowen-Margulis measure. The result is proved in the more general setting with weights coming from thermodynamic formalism, and also in the analogous setting of graphs of groups with $2$-torsion. We give new examples in real hyperbolic Coxeter groups, complex hyperbolic orbifolds and graphs of groups.
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Asymptotic growth of the number of Reciprocal Classes in the Hecke Groups
The count of reciprocal conjugacy classes in Hecke groups with word length at most x is asymptotic to a dominant geometric sum plus a normal CDF correction, with explicit parameters for each p.