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