Summer 2016

During Summer 2016 MEGL ran a program with 18 participants. There were three research groups (Orbits, Special Words, and Polytopes), two visualization groups (3D Geometric Surfaces and Virtual Reality), and one public outreach group (Outreach). The research and visualization groups engaged in experimental exploration involving faculty, graduate students, and undergraduates. Teams met weekly to conduct experiments generating data, to make conjectures from data, and to work on theory resulting from conjectures. The outreach group involved faculty, graduate students, and undergraduates to develop and implement activities for elementary and high school students that were presented at local schools and summer camps. We began with a opening BBQ, and concluded with final reports from all teams.

We are studying the dynamics of the action of outer automorphism group, $Out(F_2)$ on the character variety, $\mathfrak{X}(F_2,\mathrm{SL}_2(\mathbb{F}_q))$ for a prime $q$. One of the ways to understand the dynamics is looking at the growth of the maximum orbit length with the increase in prime $q$. We analysed the orbit growth of few outer automorphisms using Mathematica. The elements of $Out(F_2)$ exhibiting a logarithmic growth rate in the maximum orbit length is particularly interesting. Using Mathematica enabled the visualization of orbits in $\mathbb{F}_q^3$. We were also able to identify some elements of $Out(F_2)$ whose maximum orbit length is the same.
A possible future course is to determine the arithmetic ergodicity of the above action, in particular, the action of the subgroups generated by a subset of generators of $Out(F_2)$.

View our project summary here: Orbits Summary