Spherical growth of reciprocal classes in the Hecke Groups

Let $\Gamma_p$ denote the Hecke group where $p=2r$, $r>0$. Let $\mathcal{N}_l$ denote the set of conjugacy classes of reciprocal elements of word length $l$ in $\Gamma_p$. We prove that for $l \to \infty$, $$|\mathcal{N}_l| = \mathcal{O}\left(\left\lfloor \tfrac{l+1}{2} \right\rfloor^{s-1} \rho^{\left\lfloor \tfrac{l+1}{2} \right\rfloor} \right),$$ where $\mathcal O$ is the `big O', $\rho \in [\sqrt{2}, 2]$ is the unique positive real root of $$p(x) = x^{r+1} - 2\sum_{j=1}^{r-1} x^{r-j} - 1,$$ and $s$ is the maximal multiplicity among the roots of $p(x)$. Our method relies on the free product structure of the Hecke group $\Gamma_p$, a combinatorial counting function, and recurrence relations derived from cyclically reduced representatives. We also derive that the growth rate of the primitive reciprocal classes of word length $l$ is in agreement with that of $\mathcal{N}_l$. This work generalizes previous results for odd $p$ and provides an explicit asymptotic bound for all Hecke groups.