For each Hecke group Z2*Z_k, the number of primitive reciprocal conjugacy classes of word length 2t grows like a constant times the t-th power of the dominant root of an explicit polynomial, and the same growth holds for primitive classes.
Counting and equidistribution of reciprocal geodesics and dihedral groups
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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.
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Counting Reciprocal Hyperbolic Elements in Hecke Groups
For each Hecke group Z2*Z_k, the number of primitive reciprocal conjugacy classes of word length 2t grows like a constant times the t-th power of the dominant root of an explicit polynomial, and the same growth holds for primitive classes.