Combinatorial Structure of Inert and Ambiguous Classes in Modular Group

TL;DR

This paper provides new combinatorial formulas and growth rates for inert and ambiguous conjugacy classes in the modular group, offering a purely combinatorial approach distinct from previous analytic methods.

Contribution

It introduces the first combinatorial counting formulas for inert classes and develops a framework for analyzing ambiguous classes in the modular group.

Findings

Exact counting formulas for inert classes

Asymptotic growth rates for inert and ambiguous classes

A new combinatorial framework for class analysis

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.