Automorphism groups of pro-free groups and their representations

Abstract: This is joint work with Ted Chinburg and Alex Lubotzky. Let $d\ge 2$, let $F_d$ be a free discrete group of rank $d$, and let $\hat{F}_d$ be its profinite completion. Grunewald and Lubotzky developed a method to construct, under some technical conditions, representations of finite index subgroups of $\mathrm{Aut}(F_d)$ that have as images certain large arithmetic groups. In this talk, I will first show how their method leads to a stronger result for $\mathrm{Aut}(\hat{F}_d)$. I will then discuss an application of this result to Galois theory. This uses a result by Belyi who showed that there is a natural embedding of the absolute Galois group $G_{\mathbb{Q}}$ of $\overline{\mathbb{Q}}$ over $\mathbb{Q}$ into $\mathrm{Aut}(\hat{F}_2)$. In particular, I will show how the natural action of certain subgroups of $G_{\mathbb{Q}}$ on the Tate modules of generalized Jacobians of covers of $\mathbb{P}^1$ over $\overline{\mathbb{Q}}$ that are unramified outside $\{0,1,\infty\}$ can be extended, up to a finite index subgroup, to an action of a finite index subgroup of $\mathrm{Aut}(\hat{F}_2)$. If time permits, I will give a criterion for this action to define, up to a finite index subgroup, a compatible action on the Tate modules of the usual Jacobians of the covers.

combinatoricscategory theoryrings and algebrasrepresentation theory

Audience: researchers in the topic

Sherbrooke Meeting on Representation Theory of Algebras, Corona Edition (fully online)

Series comments: Please contact Thomas Brüstle or Juan Carlos Bustamante if you are interested to participate.