REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: This is joint work with Ted Chinburg and Alex Lubotzky. Let d be a fixed integer greater than 1, let F be a free discrete group of rank d, and let F^ be its profinite completion. Grunewald and Lubotzky developed a method to construct, under some technical conditions, representations of finite index subgroups of Aut(F) that have as images certain large arithmetic groups. In this talk, I will first show how their method leads to a stronger result for Aut(F^). I will then discuss an application of this result to Galois theory. This uses a result by Belyi who showed that, if d =2 , then there is a natural embedding of the absolute Galois group G_Q of the rational numbers into Aut(F^). In particular, I will show how the natural action of certain subgroups of G_Q on the Tate modules of generalized Jacobians of covers of the projective line that are unramified outside {0, 1, oo} can be extended, up to a finite index subgroup, to an action of a finite index subgroup of Aut(F^). 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.