Combinatorial Structure of Inert and Ambiguous Classes in Modular Group

arXiv (Cornell University) 2026 Preprint Open access

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Authors: Debattam Das, Krishnendu Gongopadhyay, Khushi Mishra
DOI: 10.48550/arxiv.2602.19176
Cited by: 0

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.

Fields of Study & Subjects

Advanced Algebra and Geometry Geometric and Algebraic Topology Finite Group Theory Research

Publication details

Journal / source
arXiv (Cornell University)
Publication date
22 February 2026
Language
EN
Document type
Preprint