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.
Basmajian and Mingkun Liu, Low lying geodesics on the modular surface and necklaces,Mathematical research letters, to appear
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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.