During Fall 2015 MEGL ran a 15-participant program. There were six research groups, named: Orbits, Special Words, Polytopes, Outreach, 3D printing and Virtual reality. Each of these group engaged in experimental research involving faculty, graduate students, and undergraduates. Teams met weekly to conduct experiments generating data, make conjectures from data, and work on theory resulting from conjectures. The fall semester ended with a Symposium where all teams presented their work.

Let $F_r$ denote the free group on $r$ symbols, $\mathrm{SL}(n,\mathbb{F}_q)$ is the special linear group of $n\times n$ matrices over the finite field of order $q$, and $\mathrm{Hom}(F_r, \mathrm{SL}(n,\mathbb{F}_q))$ the set of group homomorphisms from $F_r$ to $\mathrm{SL}(n,\mathbb{F}_q)$.We have been studying a particular discrete dynamical system where the state space is the character variety obtained from a geometric invariant theory quotient $\mathrm{Hom}(F_r, \mathrm{SL}(n,\mathbb{F}_q))//\mathrm{SL}(n,\mathbb{F}_q)$ (we shall denote this by $\mathfrak{X}_r(\mathrm{SL}(n,\mathbb{F}_q))$). The elements of $\mathfrak{X}_r(\mathrm{SL}(n,\mathbb{F}_q))$ are equivalence classes of homomorphisms; we will denote $[\rho]$ for the equivalence class of $\rho \in \mathrm{Hom}(F_r, \mathrm{SL}(n,\mathbb{F}_q))$. The discrete dynamical system of interest is based on the right-group action of $\mathrm{Out}(F_r)=\mathrm{Aut}(F_r)/\mathrm{Inn}(F_r)$ (which we call the outer automorphism group) on $\mathfrak{X}_r(\mathrm{SL}(n,\mathbb{F}_q))$, where $[\rho]\cdot [\alpha]=[\rho(\alpha)]$. For a fixed outer automorphism $[\alpha]$, the dynamical system has the rule of evolution $R_{[\alpha]}:\mathfrak{X}_r(\mathrm{SL}(n,\mathbb{F}_q))\times \mathbb{Z}\rightarrow \mathfrak{X}_r(\mathrm{SL}(n,\mathbb{F}_q))$ where $R_{[\alpha]}([\rho],n)=[\rho(\alpha^n)]$. Our aim is to better understand the forward orbits, i.e. the sequences $[\rho],[\rho(\alpha)],[\rho(\alpha^2)],...$ for a fixed $[\alpha]$. We started with the case of $\mathrm{SL}(2,\mathbb{F}_q))$ looking first at $F_1$ (where we studied "outer injections" or $\mathrm{Inj}(F_1)/\mathrm{Inn}(F_1)$ instead of $\mathrm{Out}(F_1)$) and are currently studying the orbits for $r=2$. The main question we have been asking is:

For a fixed $[\alpha]\in \mathrm{Out}(F_2)$, can we find a function in the order of the finite field $F(q)$ that gives the largest prime period over all elements of $\mathfrak{X}_r(\mathrm{SL}(n,\mathbb{F}_q))$? (The maximum positive integer $n$ such that there exists $[\rho]\in \mathfrak{X}_r(\mathrm{SL}(n,\mathbb{F}_q))$ with prime period $n$ under $[\alpha]$)

View the slides here: Orbits Symposium Slides