Combinatorial Structure of Inert and Ambiguous Classes in Modular Group

TL;DR

The paper addresses counting inert and ambiguous conjugacy classes in the modular group $\Gamma \cong \mathbb{Z}_2 * \mathbb{Z}_3$ using a purely combinatorial, word-length framework. By encoding conjugacy classes as AB-words and enforcing symmetry constraints from involutions, it derives exact counting formulas and sharp asymptotics for inert and ambiguous classes, including primitive/non-primitive distinctions. The main contributions are the explicit asymptotics $|\mathscr{I}_{\le 2t}| \sim \frac{2^{\lfloor t/2\rfloor}}{\lfloor t/2\rfloor+1}$ and $|\mathscr{A}_{\le 2t}| \simeq 2^{t/2}$, along with detailed combinatorial structures (anti-periodicity for inert and palindromic symmetry for ambiguous) that underpin these counts. This work provides a complementary combinatorial perspective to Sarnak's analytic results, establishing a self-contained approach to inert and ambiguous class counting in the modular group and laying groundwork for further combinatorial investigations in related symmetry classes.

Abstract

We study inert, and ambiguous conjugacy classes in the modular group $\mathrm{PSL}(2,\mathbb{Z})$ from a purely combinatorial perspective. Using word length in the free product representation $\mathbb{Z}_2 * \mathbb{Z}_3$ of the modular group, we obtain exact counting formulas and asymptotic growth rates for inert and ambiguous classes. Our results provide the first counting formulas for inert classes obtained independently of Sarnak's analytic trace-based methods, while also establishing a combinatorial framework for ambiguous classes.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Corollary 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Theorem 2.7
• Remark 2.8