We have a student seminar devoted to the problems of asymptotic group theory with some connections to ergodic theory and measure theory in general. Each talk concerns one of the problems of this beautiful area, and the thematic is considerably wide: the main idea is that somebody presents a theorem, which is proved beautifully and has a lot of applications. For example, we have proved de Finetti theorem (see Wikipedia article on it), some ergodic theorems for group actions, and Wigner's semi-circle law. I'm looking for something as beautiful and as useful, and not very difficult for being able to talk about it for 2 hours on our seminar. I hope you could give me some hints.
I repeat, that the area of problems we discuss is wide and I am not able to define it strictly, so any ideas are welcome. But generally, the methods we use are either dynamical either probabilistic. I had a thought to tell some of Grigorchuk's ideas of constructing the groups of intermideate growth, but I found those proofs technical.