Spherical growth of reciprocal classes in the Hecke Groups
Debattam Das, Krishnendu Gongopadhyay
TL;DR
The paper addresses the growth of reciprocal conjugacy classes in Hecke groups Γ_p with p=2r by exploiting the free product structure Γ_p ≅ Z_2 * Z_p and a combinatorial counting framework. It introduces a constrained composition counting function Ψ^r_n(x) and derives normal forms for reciprocal elements to obtain explicit summation formulas for class counts by type. A central result is an explicit exponential growth bound |N_l| = O( floor((l+1)/2)^{s-1} ρ^{floor((l+1)/2)} ) with ρ the unique positive root of p(x) = x^{r+1} - 2∑_{j=1}^{r-1} x^{r-j} - 1, and s the maximal root multiplicity; this bound holds for all Hecke groups, including even p, and extends prior odd-p results. The authors also show that the growth rate of primitive reciprocal classes matches that of all reciprocal classes and provide detailed, type-specific counting formulas, offering a comprehensive asymptotic picture and explicit combinatorial counts across all p.
Abstract
Let $Γ_p$ 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 $Γ_p$. We prove that for $l \to \infty$, $$|\mathcal{N}_l| = \mathcal{O}\left(\left\lfloor \tfrac{l+1}{2} \right\rfloor^{s-1} ρ^{\left\lfloor \tfrac{l+1}{2} \right\rfloor} \right), $$ where $\mathcal O$ is the `big O', $ρ\in [\sqrt{2}, 2]$ 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 $Γ_p$, 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.