Spherical growth of reciprocal classes in the Hecke Groups

Abstract

Let $r\in \mathbb{Z}$, $r>2$, and let ${\Gamma }_{2r}\cong (\mathbb{Z}/2\mathbb{Z})\ast (\mathbb{Z}/2r\mathbb{Z})$ be the Hecke group. Let ${\mathcal{N}}_{\ell }$ denote the set of conjugacy classes of reciprocal elements with cyclically reduced word length exactly $\ell$ in the standard generating set of ${\Gamma }_{2r}$. Let $P(x)$ be the polynomial \[ P(x) = x^{r+1} - 2\sum_{i=1}^{r-1} x^{i} - 1. \] Let $\rho$ be the unique positive root of $P$ and $C$ be a constant depending on $r.$ We prove that, as $\ell \to \infty$, \[|\mathcal{N}_\ell| \sim Cρ^{\left\lfloor \tfrac{\ell+1}{2} \right\rfloor}. \] Our approach exploits the free product structure of ${\Gamma }_{2r}$, together with a combinatorial counting function and recurrence relations arising from cyclically reduced representatives. We further show that the asymptotic growth rate of primitive reciprocal conjugacy classes of word length $\ell$ coincides with that of ${\mathcal{N}}_{\ell }$. This result extends previous work for odd values of $p$ and provides a uniform asymptotic bound for all Hecke groups.

arXiv commentto be published in Illinois Journal of Mathematics