250059 VO Geometric and asymptotic group theory (2011W)

The course is on infinite random groups. These are groups obtained using a random choice of group relators. There are various models of random groups: combinatorial, topological, statistical, etc. The idea goes back to works of Gromov and Ol'shanskii.

We will give an elementary account of the subject. First we introduce basic notions of geometric and asymptotic group theory such as van Kampen diagrams and Dehn's isoperimetric functions. Then we will proceed with a short discussion of small cancellation theory and Gromov's hyperbolic groups, and give a combinatorial proof of Gromov's small cancellation theorem stating that a graphical small cancellation group is hyperbolic.

The main technical goal we pursue is Gromov's sharp phase transition theorem: a random quotient of the free group F_m is trivial in density greater than 1/2, and non-elementary hyperbolic in density smaller than this value. This refers to the density model of random groups, where the choice of group relators depends on the density parameter d with values between 0 and 1.