Looking for at least one beautiful and not too technical result in asymptotic group theory 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.
 A: You might get a good seminar talk out of certain cases of the Patterson-Sullivan theory, for instance the original case considered by Patterson for convex cocompact actions of a finite rank free group on the hyperbolic plane. You get a nice formula for the Hausdorff dimension of its Cantor set at infinity. The formula is expressed by using the asymptotics of the orbits in the hyperbolic plane to construct certain probability measures on the Cantor set (known nowadays as the Patterson measure or in a more general context as Patterson-Sullivan measures), and then understanding the Radon-Nykodym derivatives of the group action on these measures.
A: Wolfgang Lueck has written a survey 
http://arxiv.org/abs/0806.3771
Theorem 1.1 (Schreier’s Theorem)
Let $G$ be a free group and $H \subset G$ be a
subgroup. Then $H$ is free. If the rank $rk(G)$ and the index $[G : H]$ are ﬁnite,
then the rank of $H$ is ﬁnite and satisﬁes
$$rk(H) = [G : H] · (rk(G) − 1) + 1.$$
A: You can speak about the Howe-Moore theorem, which is very useful and imply ergodicity (and actually, mixing) of group actions on reasonable spaces.
