Counting Reciprocal Hyperbolic Elements in Hecke Groups

A reciprocal geodesic on a (2,k, $\infty$) Hecke surface is a geodesic loop based at an even order cone point p traversing its path an even number of times. Associated to each reciprocal geodesic is the conjugacy class of a hyperbolic element in the (2,k,$\infty$) Hecke group whose axis passes through a cone point that projects to p. Such an element is called a reciprocal hyperbolic element based at p. In this paper, we determine the asymptotic growth rate and limiting constant (in terms of word length) of the number of primitive conjugacy classes of reciprocal hyperbolic elements in a Hecke group.