Randomizations were introduced by Keisler and Ben Yaacov and they can be understood intuitively as random variables with values in M. In this talk we will give a brief introduction to the subject and study two kinds of groups that appear naturally in the construction:
1. When M is a group, the randomization inherits a natural pointwise group operations that inherits many properties from M: being abelian, definably nilpotent, etc. We show (joint work with Muñoz) that when T is stable its randomization is always connected group.
2. The group of isometries of these structures have been characterized and studied by Ibarlucía. They can be understood in terms of the group of automorphisms of M. We will discuss several topologies that arise naturally in this group and prove (joint work with Zamora) how some dynamical properties of Aut(M) transfer to the group of isometries of its Borel randomization. | 
