Freedom from within : seeking generating sets for free groups

Since the 1870's, mathematicians have had only one real tool to help identify when a group contains free subgroups: the Klein Criterion. While powerful and elegant, there are cases when this theorem does not prove helpful. In order to apply it, we must have a candidate set that could generate a free group. This is no longer the case. The Sunic Criterion is a new theorem discovered by Zoran Sunic in 2013, and it does not require us to have a candidate for a generating set beforehand! In theory, this could allow mathematicians access to a host of groups which until now have been out of reach. In this thesis we use this new theorem to reprove that certain groups contain free subgroups and expand upon it showing that it can also be used to nd explicit generating sets for locally free groups. Having done this, we nd explicit generat- ing sets for locally free subgroups in PSL2(Z) and Aut(R+): We also explore the conditions of this new theorem.