  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.
