Abstract:

When working over a group instead of a field, the analogue of a polynomial is just a word w(x_1, ..., x_n) which can involve constants from the group. Following this analogy, one can study varieties, and then algebraic groups (varieties endowed with a group law which can be expressed by such a "polynomial").
Over the free group, varieties are well understood since the works of Makanin-Razborov and later Sela. In a joint work with Guirardel, we show that there are very few irreducible algebraic groups over the free group, and we describe all such structures. One of the key tools in this work is JSJ decomposition of groups, which enables us to give a description of automorphisms of the coordinate group of such a variety in terms of automorphisms of fundamental groups with boundaries.