Lior Rosenzweig: Galois groups of random elements of linear groups

Let A be a finitely generated subgroup of GL_n(k), where k is a finitely generated field of characteristic zero. In the talk we will discuss what type of groups can occur as Gal(k(g)/k), where g is an element of A, and k(g) is the splitting field of the characteristic polynomial of g. In particular, we will show that if the Zariski closure of A does not contain a central torus (e.g semisimple), then given a random walk on A, the behaviour of Gal(k(g)/k) is generic with respect to connected components of the Zariski closure. The proof uses the recently developed "sieve theory for groups" via expenders and property 'tau' of linear groups.