At this point, we’ve successfully created an algorithm that can find valid generalized ping-pong intervals in \(\mathbb{RP}^1\) for a representation of certain strongly geodesically automatic into \(SL(2,\mathbb{R})\), and have the foundations of a method for computing an upper bound on the size of the kernel of this representation. However, there are still a few unfinished pieces to this project.

Exploring Image Patching

Because containment rules for generalized ping-pong don’t require that intervals are disjoint or connected, we implemented “image patching” as a search algorithm to find valid intervals. While image patching is much faster than linear expansion and seems generally promising, it still doesn’t fully work for the triangle group. Moreover, as we test more complex groups, the intervals become difficult to visualize with our current program’s output. In order to resolve the current issues with image patching, we’ll need to first rework our visualizations to investigate what the specific problems are, and then we can work on finding a resolution to ensure that image patching functions correctly. 

Explicit Upper Bound for Size of the Kernel

Once the program has found valid intervals, it guarantees that there is some finite upper bound on the size of the kernel. As previously discussed, finding this upper requires calculating two values: \(\lambda\) and \(C\). We’ve mostly developed a method to find \(\lambda\), but there are still a few steps of calculus that we haven’t yet implemented into the program. As for \(C\), while it should be theoretically easier to calculate, we haven’t done much work on explicitly finding this value. Once \(\lambda\) and \(C\) are found, the upper bound can be calculated, which will determine the faithfulness of the representation.

Extending the Algorithm to \(\mathbf{SL(n,\mathbb{R})}\) and \(\mathbb{RP}^{n-1}\)

Currently, the algorithm supposes that the representation is into \(SL(2,\mathbb{R})\), and acts on \(\mathbb{RP}^1\). However, we should be able to extend it to generalized \(n\). This would mean adjusting the algorithm so it searches for intervals in \(\mathbb{RP}^{n-1}\), which would verify the faithfulness of a representation into \(SL(n,\mathbb{R})\). For example, in the case \(n = 3\), we’d work with a representation into \(SL(3,\mathbb{R})\) and search for valid regions in \(\mathbb{RP}^2\). Perhaps this will be “better” in some sense for finding intervals for certain groups.