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

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.