Spherical growth of reciprocal classes in the Hecke Groups

TL;DR

This paper establishes the asymptotic growth rate of reciprocal conjugacy classes in Hecke groups, generalizing previous results and providing explicit bounds based on algebraic and combinatorial methods.

Contribution

It introduces a new asymptotic bound for reciprocal classes in all Hecke groups, extending prior work limited to odd p, using recurrence relations and group structure.

Findings

Growth rate of reciprocal classes is exponential with respect to word length.

Explicit asymptotic bounds depend on roots of a specific polynomial.

Results unify and extend previous bounds for different p values.

Abstract

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.