** Geometric and asymptotic group theory I **

Random groups

Lecture by Goulnara Arzhantseva

and

Problem session by Damian Osajda

Dienstag, 10:00--12:00, Raum C2.07 UZA 4

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 0< d <1.

Course assessment: Presentation or test.

The first hour (10:00--10:45) is a lecture, and the second hour (11:00--11:45) is a problem session.

Lists of problems:
__Blatt 1__, __Blatt 2__,
__Blatt 3__, __Blatt 4__,
__Blatt 5__, __Blatt 6__

1st EXAM. January 31 from 17:00 till 18:00, room D107. Note: 20 min preparation, 20 min oral presentation (time is strict !), one A4 format page of "reminder" is allowed.

**2nd EXAM.** March 13 from 17:00 till 18:00, room C207. Note: 20 min preparation, 20 min oral presentation (time is strict !), one A4 format page of "reminder" is allowed.

**References **(advised but not obligatory!):

- Free groups, group presentations:

Ch. 2 (pages 52-56, 58-60, 71-74, 81-84) of the book
by Oleg Bogopolski, Introduction to group theory

Chapters 1 and 4 of the book by.D.L. Johnson, Presentations of Groups (second edition)

- Van Kampen diagrams:

Hamish Short, Introduction to the geometry of the word problem in
"The geometry of the word problem for finitely generated groups", Advanced Courses in Mathematics - CRM Barcelona, 2006.

- Graphical small cancellation:

Yann Ollivier, On a small cancellation theorem of Gromov, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 1, 75-89.

- Phase transition theorem:

Ch.2, pages 607-615 of Yann Ollivier, Sharp phase transition theorems for hyperbolicity of random groups, GAFA 14 (2004), available at
http://www.yann-ollivier.org/rech/publs/quothyp_publ.pdf