February 6, 2023

## On representations of automorphism groups of free profinite groups and of the absolute Galois group over Q

Frauke Bleher, University of Iowa, USA Zoom only · JP:21:00 · FR:13:00

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 $$\mathrm{Aut}(\hat{F}_2)$$. I will also 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.