In this thesis, we study finitely generated subgroups of the matrix group SL₂ (over various locally compact fields) which are both discrete and free.

We first examine the existing literature on two- and three-generated subgroups of SL₂(R) and SL₂(C). Some such subgroups are known to be free, and this can be proved by applying a ‘combination’ theorem (such as Klein’s Combination Theorem, or the Ping Pong Lemma) to the action of these groups by Möbius transformations on the Riemann sphere. It remains, however, an open problem to determine freeness of such subgroups in general. On the other hand, applying the Ping Pong Lemma to the action of SL₂(R) by Möbius transformations on the hyperbolic plane H² is known to give necessary and sufficient conditions for a two-generated subgroup of SL₂(R) to be both discrete (with respect to the topology inherited from R⁴) and free of rank two. This forms the basis of an existing practical algorithm which, given a two-generated subgroup G ≤ SL₂(R), determines after finitely many steps whether or not G is both discrete and free of rank two.

We then look at two-generated subgroups of SL₂(K), where K is a non-archimedean local field (such as the p-adic numbers). Such groups act by isometries and without inversions on a locally finite regular simplicial tree, called the Bruhat-Tits tree. We demonstrate that applying the Ping Pong Lemma to this action gives a practical algorithm which, given a two-generated subgroup G ≤ SL₂(K), determines after finitely many steps whether or not G is both discrete (with respect to the topology inherited from K⁴) and free of rank two. The basis of this algorithm involves computing and comparing various translation lengths.

Finally, we show that similar techniques can be used to give another algorithm which, given a three-generated subgroup G ≤ SL₂(K), determines after finitely many steps whether or not G is both discrete and free of rank three. We demonstrate that both algorithms can be applied more generally in the setting of two- or three-generated subgroups of the isometry group of any locally finite simplicial tree (when equipped with the topology of pointwise convergence, and a method of computing translation lengths) and have relevance to the constructive membership problem.