On the Local-Global Conjecture for Commutator Traces

TL;DR

This paper investigates the trace set of the commutator subgroup of , identifying local and global obstructions, and develops probabilistic and algorithmic methods to understand trace representations, proposing a conjecture on minimal commutator counts.

Contribution

It introduces a new approach combining local-global analysis, probabilistic modeling, and algorithms to study traces in the commutator subgroup of , and formulates a conjecture on minimal commutator representations.

Findings

Identified local obstructions to trace sets.

Developed a probabilistic model for trace existence.

Proposed that traces can be represented with 1 or 2 commutators.

Abstract

We study the trace set of the commutator subgroup of $\Gamma(2),$ a type of Local-Global problem about thin groups. We determine the local obstructions and then use the correspondence between binary quadratic forms and hyperbolic matrices to find some global obstructions. We then develop a probabilistic argument for the existence of sufficiently large admissible traces by modeling the elements as non-backtracking random walks in a 2-dimensional lattice via their homology class and word length. Finally, we investigate the number of commutators needed to represent matrices in the commutator subgroup. This is done using an algorithm of Goldstein and Turner along with utilizing properties of level $k$-Markoff type surfaces. We conjecture that any trace in the commutator subgroup of $\Gamma(2)$ can be represented with either 1 or 2 commutators.