Is it true that if many elements of a group commute, then the group is commutative? What happens if most of the elements have order 2? Can every group be realised as the automorphism group of a "nice" graph? These questions have been solved in the 70-90's for finite groups. But what happens for infinite groups? How can we interpret "many elements" or "most of the elements" for infinite groups?

In this talk we will see how we can use random walks to solve the above questions for infinite (finitely generated) groups.

The talk is supposed to be a gentle introduction to random walks and infinite group theory; showing how probability theory and algebra nicely interact. No prerequisites are required and the talk will accessible to second year bachelor students.