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250110 VO Random Groups (2020W)

3.00 ECTS (2.00 SWS), SPL 25 - Mathematik



Language: English

Examination dates


Classes (iCal) - next class is marked with N

Tuesday 06.10. 09:45 - 12:15 Digital
Tuesday 13.10. 09:45 - 12:15 Digital
Tuesday 20.10. 09:45 - 12:15 Digital
Tuesday 27.10. 09:45 - 12:15 Digital
Tuesday 03.11. 09:45 - 12:15 Digital
Tuesday 10.11. 09:45 - 12:15 Digital
Tuesday 17.11. 09:45 - 12:15 Digital
Tuesday 24.11. 09:45 - 12:15 Digital
Tuesday 01.12. 09:45 - 12:15 Digital
Tuesday 15.12. 09:45 - 12:15 Digital
Tuesday 12.01. 09:45 - 12:15 Digital
Tuesday 19.01. 09:45 - 12:15 Digital
Tuesday 26.01. 09:45 - 12:15 Digital


Aims, contents and method of the course

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.

Assessment and permitted materials

Oral exam or written manuscript. The choice is to make at the beginning of the course.

Minimum requirements and assessment criteria

The knowledge of basic concepts in algebra, topology and probability is required (examples are groups, fundamental group, group action, planar map, probabilistic pigeonhole principle, etc.).

Examination topics

Content of the lectures and exercises.

Reading list

Association in the course directory


Last modified: Tu 20.04.2021 16:48