Combinatorial Structure of Inert and Ambiguous Classes in Modular Group

We study inert, and ambiguous conjugacy classes in the modular group $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{Z})$ from a purely combinatorial perspective. Using word length in the free product representation ${\mathbb{Z}}_2\ast {\mathbb{Z}}_3$ 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.