Although the project we set out to complete is nearing its end, there are some clear places to extend the work for potential future projects. We'll list a few ideas for projects here as well as some ways to go about starting them.

Extending to \(\mathbb{RP}^2\) for representations in \(SL(3,\mathbb{R})\)

We've mentioned this idea for a future project before, but on completing this algorithm for \(SL(2,\mathbb{R})\) and \(\mathbb{RP}^1\), we now suggest a smaller step up a single dimension instead of generalizing immediately to n dimensions. The patching algorithm is likely to work here, but some work needs to be done to define intervals in this higher dimension. Up until now, intervals of \(\mathbb{RP}^1\) have been easy to define by two endpoints, but going up a dimensions, intervals will become disconnected regions of the plane. Our suggestion for starting a project in this direction would be to replace initial intervals with small triangular regions around singular directions of \(SL(3,\mathbb{R})\) matrices and combining regions by taking the convex hull of the union of their vertices.

Robustness for a wider range of representations

Towards the end of the project, we found some examples of representations of surface groups of a different presentation which the algorithm seemed to struggle with. The issue here is likely due both to the highly compressive matrices which represent the group generators and the close proximity that their fixed points lie in on \(\mathbb{RP}^1\). As we write this, we plan to spend a bit more time searching for a fix to this issue, but our plan is to use fewer initial intervals, each around singular directions of much longer words than we're using now. This should hopefully decrease the chance of intervals starting where they shouldn't and also improve the speed of the algorithm.