Spherical growth of reciprocal classes in the Hecke Groups

Let ${\mathrm{\Gamma }}_p$ 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 ${\mathrm{\Gamma }}_p$. We prove that for $l\to \mathrm{\infty }$, $$\mathrm{\mid }{\mathcal{N}}_l\mathrm{\mid }=\mathcal{O}\left({\lfloor \frac{l+1}{2}\rfloor }^{s-1}{\rho }^{\lfloor \frac{l+1}{2}\rfloor }\right),$$ 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 ${\mathrm{\Gamma }}_p$, 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.