Programs & Pictures

As a starting point for this paper I made a program which draws the limit curves of SL(3,R) triangle group representations in RP². These can be convex / Hitchin representations, but also Barbot-type ones like in the picture on the left. These have a really interesting fractal structure, and we could guess many of their weird properties by just staring at these pictures for a long time.
The above program can not only make still pictures or interactively let's you change the parameters, but you can also generate animations. A really nice example is this one. The three lines and points define three projective reflections in RP², which together generate a (5,5,5) triangle group. Up to projective equivalence there is only a one-dimensional space of such groups, and the animation progresses through this parameter. Then we draw the attracting fixed point of the product of the three reflections, and its orbit by a few thousand group elements. You clearly see two distinct behaviours: when the representation is Anosov, the orbit lies on a fractal curve, the limit curve. Otherwise, it's just randomly distributed through the plane, but shows some fascinating patterns. Click the picture to see the animation.
Together with David Dumas I made a program to enumerate balanced ideals in Weyl groups. These are combinatorial objects which allow you to construct cocompact domains of discontinuity in flag manifolds for Anosov representations, and (at least in some cases) classify these domains.
This is what you get when you take a hyperbolic triangle rotation group in SL(3,R) and ask which element maximizes what we call the "slope of the Jordan projection", the quotient of the logarithms of the highest and lowest eigenvalues. The two-dimensional picture represents the two-dimensional space of such groups (actually just a quarter of it), and the colors are different maximizing elements (or conjugacy classes, if you want to be more precise). Where does this foliation-like structure come from? I'm still exploring that with Jeff Danciger, but it's certain that there's some interesting geometry involved here!