Recent results have now given a more or less complete picture of the parabolic representation space of F2 in PSL(2,C). This classification problem splits into two parts - the discrete and non-discrete representations. The classification for the former starts with work of Shimitzu-Leutbecher and Lyndon-Ullman in the late 1960's and is completed by various important conjectures (ending lamination, density, model manifolds, etc) for Kleinian groups and the recent solution of Agol's conjecture.For non-discrete representations very little was known until my recent results showing supergroup density and the existence of arbitrarily large Nielsen classes.   These last results are proved through studying the action of a polynomial semigroup on the character variety of F2.  Ultimately it’s a story about two matrices,  how very different discrete and non-discrete groups are when they are not free,  and it will be revealed at an elementary level.