Unlike the ping pong lemma, finding valid intervals for generalized ping pong leaves us with a bit more work to do. As covered in the Generalized Ping Pong section, we are left with a bound on the size of the kernel instead of knowledge that that kernel is just 0. After finding valid intervals, we can theoretically compute what this upper bound is and then simply check all words of length up to a number related to that bound in order to officially verify the faithfulness of the representation.
The paper ‘Uniform Hyperbolic Finite-Valued SL(2,R)-Cocycles‘ outlines this bound in Chapter 2 but does not give a means of explictly computing it. In order to get a numerical bound as a function of our valid intervals, we need to solve for two separate values: λλ and CC. We’ve found this to be quite a lot of work to do, so we’ve finished some of the calculation for λλ and left finding CC for future work.
