Title: Galois group of random elements of linear groups

Abstract:
Let A be a finitely generated subgroup of GLn(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
is 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.
This is a joint work with Alex Lubotzky