On the local-global conjecture for commutator traces

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