Geometric Group Theory, SS 2022

News

Exercise sheet 14 is online.
Help Blorx!
26.07.2022: Problem 1 is corrected.
The evaluation feedback can be found on GRIPS. Thanks for participating!
Exams: oral exams (ca. 25 minutes)
Exam dates: 01.08.2022; 15.09.2022; 13.10.2022;
Registration: Step 1: register for a time slot with Ms Bonn (you can visit her in person in M217 or send an email to rosina.bonn@mathematik.uni-r.de). Step 2: register in FlexNow.
Format: as long as the pandemic rules allow for it, you can choose between an in-person and an online exam.
Contents: I do not expect you to know any details of covering theory or Riemannian geometry; but you should be aware of the existence and rough shape of the applications to Riemannian geometry.
Exercise sheet 7 is online.
Lean src for the bonus problem: quasiisometry_exercise.lean
07.06.2022: reformulated Quick check C.
09.06.2022: ... and (in the same way ...) Exercise 1.
Organisational matters; in particular: registration for the exercise classes
The Seminar Thompson Groups complements this course.
The organisational meeting is on Thu, Feb 10, 13:00, online (zoom access data will be available through GRIPS)
If you plan to write a bachelor thesis under my supervision in WS 2022/23 (in Topology/Geometry), you should participate in a seminar in the Global Analysis and Geometry group before WS 2022/23.

Geometric Group Theory

Geometric Group Theory connects geometric and algebraic properties of groups. Typical questions are:

Can groups be viewed as geometric objects and how are algebraic properties related to geometric properties?
More generally: On which geometric objects can specific groups act in a reasonable way and how are properties of such objects/actions related to algebraic properties?

For instance, freeness of groups can be characterised in terms of actions on trees. This gives a proof of the fact that subgroups of free groups are free. In this course, we will translate geometric notions such as geodesics, curvature, volume, etc. into the world of group theory. The topics will be adapted to the background and interests of the participants

If all participants agree, this course can be held in German; solutions to the exercises can be handed in in German or English.

Lecture notes:

pdf
The lecture notes are based on the book C. Löh. Geometric Group Theory. An Introduction. Universitext, Springer, 2017.
Free draft of the book: pdf
Errata of the book: pdf

Topics covered so far:

Literature
Introduction
Generating Groups
- Review of the category of groups
- Groups via generators and relations
  [Generating sets of groups, Free groups, Generators and relations, Finitely presented groups]
- New groups out of old
  [Products and extensions, Free groups and amalgamated free groups]
Cayley Graphs
- Basic notions from graph theory
- Cayley graphs
- Cayley graphs of free groups
  [Free groups via reduced words, Free groups -> trees]
Group Actions
- Review of group actions
  [Free actions, Orbits and stabilisers, Transitive actions]
- Free groups and actions on trees
  [Spanning trees for group actions, Reconstructing a Cayley tree, Applications: Subgroups of free groups are free]
- The ping-pong lemma
  [Application: The group SL(2,\Z) is virtually free, Application: Regular graphs of large girth, Application: The Tits alternative]
Quasi-isometry
- Quasi-isometry types of metric spaces
- Quasi-isometry types of groups
  [First examples]
- Quasi-geodesics and quasi-geodesic spaces
  [(Quasi-)geodesic spaces, Geodesification via geometric realisation of graphs]
- The \v Svarc-Milnor lemma
  [Application: (Weak) commensurability, Application: Geometric structures on manifolds]
- The dynamic criterion for quasi-isometry
- Quasi-isometry invariants
Growth types of groups
- Growth functions of finitely generated groups
- Growth types of groups
  [Growth types, Growth types and quasi-isometry, Application: Volume growth of manifolds]
- Groups of polynomial growth
  [Nilpotent groups, Growth of nilpotent groups, Polynomial growth implies virtual nilpotence, Applications: Virtual nilpotence, More on polynomial growth]
- Groups of uniform exponential growth
  [Uniform exponential growth, Uniform uniform exponential growth, Application: The Lehmer conjecture]
Hyperbolic groups
- (Quasi-)Hyperbolic spaces
  [Hyperbolic spaces, Quasi-hyperbolic spaces, Quasi-geodesics in hyperbolic spaces, Hyperbolic graphs]
- Hyperbolic groups
- Solving the word problem in hyperbolic groups
- Elements of infinite order in hyperbolic groups
  [Existence, Centralisers, Quasi-convexity]
The End.
Appendix
- Quasi-isometries in a proof assistant
  source code: quasiisometry.lean
  additional files: leanpkg.toml leanpkg.path LICENSE.txt
  Lean web editor

Time/Location

Tuesday, 8:30--10, M 104
Fridday, 8:30--10, M 104

On demand, we can switch to a hybrid format (with zoom live streaming); please contact Clara Löh by email. The access data is available on GRIPS.

Exercise classes

Time/Location: tba (probably: Mo 8:30--10, M 102)

Exercise sheets

Solutions can be submitted in English or German via GRIPS and in teams of up to two people. Please do not forget to add your name to all your submissions!

Problems marked as 'Quick check' are not to be submitted and will not be graded; these problems will be discussed in the exercise classes.

Sheet 1,	of April 26, 2022,	submission before May 3, 2022 (08:30)	will be discussed in the exercise classes in the second/third week
Sheet 2,	of May 3, 2022,	submission before May 10, 2022 (08:30)	will be discussed in the exercise classes in the 3rd/4th week
Sheet 3,	of May 10, 2022,	submission before May 17, 2022 (08:30)	will be discussed in the exercise classes in the 4th/5th week
Sheet 4,	of May 17, 2022,	submission before May 24, 2022 (08:30)	will be discussed in the exercise classes in the 5th/6th week
Sheet 5,	of May 24, 2022,	submission before May 31, 2022 (08:30)	will be discussed in the exercise classes in the 6th/7th week
Sheet 6,	of May 31, 2022,	submission before June 8(!), 2022 (08:30)	will be discussed in the exercise classes in the 7th/8th week
Sheet 7,	of June 7, 2022,	submission before June 14, 2022 (08:30)	will be discussed in the exercise classes in the 8th/9th week
			quasiisometry_exercise.lean
Sheet 8,	of June 14, 2022,	submission before June 21, 2022 (08:30)	will be discussed in the exercise classes in the 9th/10th week
Sheet 9,	of June 21, 2022,	submission before June 28, 2022 (08:30)	will be discussed in the exercise classes in the 10th/11th week
Sheet 10,	of June 28, 2022,	submission before July 5, 2022 (08:30)	will be discussed in the exercise classes in the 11th/12th week
Sheet 11,	of July 5, 2022,	submission before July 12, 2022 (08:30)	will be discussed in the exercise classes in the 12th/13th week
Sheet 12,	of July 12, 2022,	submission before July 19, 2022 (08:30)	will be discussed in the exercise classes in the 13th/14th week
Sheet 13,	of July 19, 2022,	optional submission before July 26, 2022 (08:30)	the quick checks will be discussed in the exercise classes in the 14th week
Sheet 14,	of July 26, 2022,	no submission	Help Blorx!

Literature

A guide to the literature will be provided at the beginning of SS 2022.

Prerequisites

All participants should have a firm background in Analysis I/II (in particular, basic point set topology, e.g., as in Analysis II in WS 2011/12), in Linear Algebra I/II, and basic knowledge in group theory (as covered in the lectures on Algebra).
Experience in Geometry/Topology is helpful, but not necessary.

Exams

Please read the information on organisation and formalities of this course.

Last change: July 29, 2022