This course — known as the Berstein Seminar — is a geometry/topology seminar in which the students do most of the lecturing and the professor tries to provide guidance without getting in the way.
Following the tradition of the seminar, the students will be part of the decision about the topic. My suggestions are:
Groups acting on trees. There is a lot of interesting material about this. Bass-Serre theory on groups acting on simplicial trees and relates to ways of splitting groups as free and amalgamated products.
Arithmetic groups and lattices. This is a large area — we will focus mainly on lattices in p-adic Lie groups or in the case of positive characteristic.