Universität Wien
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250122 VO Random Groups (2022W)

3.00 ECTS (2.00 SWS), SPL 25 - Mathematik
ON-SITE
Moodle

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Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

  • Tuesday 11.10. 09:45 - 12:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 18.10. 09:45 - 12:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 25.10. 09:45 - 12:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 08.11. 09:45 - 12:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 15.11. 09:45 - 12:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 22.11. 09:45 - 12:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 29.11. 09:45 - 12:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 06.12. 09:45 - 12:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 13.12. 09:45 - 12:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 10.01. 09:45 - 12:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 17.01. 09:45 - 12:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 24.01. 09:45 - 12:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 31.01. 09:45 - 12:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock

Information

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

An oral exam or a written manuscript. The choice is to make at the beginning of the course.

Minimum requirements and assessment criteria

The knowledge of very basic concepts in algebra, topology and probability is required: examples are graphs, groups, group action, probability of an event, etc.

Examination topics

Content of the lectures and exercises.

Reading list

Slides of lectures will be available on the Moodle.

Association in the course directory

MALV

Last modified: Mo 15.05.2023 14:48