Introduction
In this project, we focus on extending the work done last semester by designing an algorithm around a generalized version of the Ping Pong Lemma to work for a larger set of groups. The previous project allowed us to find intervals of \(\mathbb{RP}^1\) which met certain containment conditions, guaranteeing the faithfulness of particular representations of free groups in \(\text{SL}(2,\mathbb{R})\). Here, we attempt to generalize these conditions to work for any group with an automatic structure. This will allow us to find valid intervals under a different set of conditions to work for non-free groups such as cyclic free products and triangle groups.