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250078 VO Random Groups (2025W)
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Note: The time of your registration within the registration period has no effect on the allocation of places (no first come, first served).
Details
Language: English
Lecturers
Classes (iCal) - next class is marked with N
- N Tuesday 14.10. 09:45 - 11:15 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
- Tuesday 21.10. 09:45 - 11:15 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
- Tuesday 28.10. 09:45 - 11:15 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
- Tuesday 04.11. 09:45 - 11:15 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
- Tuesday 11.11. 09:45 - 11:15 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
- Tuesday 18.11. 09:45 - 11:15 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
- Tuesday 25.11. 09:45 - 11:15 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
- Tuesday 02.12. 09:45 - 11:15 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
- Tuesday 09.12. 09:45 - 11:15 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
- Tuesday 16.12. 09:45 - 11:15 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
- Tuesday 13.01. 09:45 - 11:15 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
- Tuesday 20.01. 09:45 - 11:15 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
- Tuesday 27.01. 09:45 - 11:15 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
Information
Aims, contents and method of the course
Assessment and permitted materials
An oral exam.No any course or exercises material can be used during the exam.The exam process:
You enter the room at the exact time (which you receive by email) and you choose randomly one question from the list of exam questions.
You have 30 minutes to preparer your answer (only empty piece of paper and a pen is used during the preparation).
Then you have 25 minutes to present your answer orally and on the board. I can ask questions during your presentation.
Then I ask an additional question (on a topic different from your chosen question).
The grade will appear on the u:space in the coming days.
You enter the room at the exact time (which you receive by email) and you choose randomly one question from the list of exam questions.
You have 30 minutes to preparer your answer (only empty piece of paper and a pen is used during the preparation).
Then you have 25 minutes to present your answer orally and on the board. I can ask questions during your presentation.
Then I ask an additional question (on a topic different from your chosen question).
The grade will appear on the u:space in the coming days.
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
Exam questions:(1) Free groups: definition and (non)-examples, the Cayley graph, rank, Sanov’s theorem.
(2) Graphs representing finitely generated subgroups of free groups: definition, examples,
properties, Stallings theorem.
(3) Group presentations: definition, examples, description of the kernel of the canonical
epimorphism Fm—>G (this is (i) <=>(ii) in the van Kampen Lemma).
(4) Finitely presented groups: definition, examples, von Dyck’s theorem.
(5) The few relator model of random groups: definition and examples of generic properties
of finitely presented groups.
(6) Classical small cancellation condition C’(1/6): definition, (non)-examples, geometric
interpretation.
(7) A random group in the few relator model satisfies classical C’(µ), for any given µ>0.
(8) The “big face” theorem (with proof).
(9) Van Kampen diagrams: definition, examples, van Kampen lemma.
(10) Word problem and Dehn algorithm: formulation, examples, proof in the C’(1/6) case.
(11) Isoperimetric inequality and Dehn functions: definition, examples, theorem on
recursive Dehn functions.
(12) Gromov’s density model of random groups: definition and examples of random
properties.
(13) The probabilistic pigeon-hole principal (with proof).
(14) Gromov’s phase-transition theorem: formulation and proof of case d > 1/2.
(15) Gromov’s phase-transition theorem: proof of the case when all relators are different.
(16) The space of marked groups: definition and topological properties.
(17) The space of marked groups: examples of converging sequences and topological
genericity.
(18) Graphical small cancellation conditions: definition and examples.
(19) Graphical small cancellation condition Gr’(1/6): isometric embedding theorem.
(20) Graph coverings, graphical quotients from graph coverings.
(21) Graphical small cancellation via graph coverings.
(22) Small cancellation guarantees the infinity of the group: examples.
(23) Large girth dg-bouded graphs: definition and examples. Proof of logarithmic girth of Margulis’ graph.
(24) Construction of exceptional groups: strategy, the use of expanders and of graph
coverings.
(2) Graphs representing finitely generated subgroups of free groups: definition, examples,
properties, Stallings theorem.
(3) Group presentations: definition, examples, description of the kernel of the canonical
epimorphism Fm—>G (this is (i) <=>(ii) in the van Kampen Lemma).
(4) Finitely presented groups: definition, examples, von Dyck’s theorem.
(5) The few relator model of random groups: definition and examples of generic properties
of finitely presented groups.
(6) Classical small cancellation condition C’(1/6): definition, (non)-examples, geometric
interpretation.
(7) A random group in the few relator model satisfies classical C’(µ), for any given µ>0.
(8) The “big face” theorem (with proof).
(9) Van Kampen diagrams: definition, examples, van Kampen lemma.
(10) Word problem and Dehn algorithm: formulation, examples, proof in the C’(1/6) case.
(11) Isoperimetric inequality and Dehn functions: definition, examples, theorem on
recursive Dehn functions.
(12) Gromov’s density model of random groups: definition and examples of random
properties.
(13) The probabilistic pigeon-hole principal (with proof).
(14) Gromov’s phase-transition theorem: formulation and proof of case d > 1/2.
(15) Gromov’s phase-transition theorem: proof of the case when all relators are different.
(16) The space of marked groups: definition and topological properties.
(17) The space of marked groups: examples of converging sequences and topological
genericity.
(18) Graphical small cancellation conditions: definition and examples.
(19) Graphical small cancellation condition Gr’(1/6): isometric embedding theorem.
(20) Graph coverings, graphical quotients from graph coverings.
(21) Graphical small cancellation via graph coverings.
(22) Small cancellation guarantees the infinity of the group: examples.
(23) Large girth dg-bouded graphs: definition and examples. Proof of logarithmic girth of Margulis’ graph.
(24) Construction of exceptional groups: strategy, the use of expanders and of graph
coverings.
Reading list
Slides of lectures will be available on the Moodle.
Association in the course directory
MALV; ML2; MEL
Last modified: Mo 28.07.2025 11:26
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.