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Ergodic Theory of Groups, SS 2020

Prof. Dr. C. Löh / J. Witzig

News

• The lecture notes are updated (29.07.2020); correction log.
• All handwritten notes in a single file: pdf (22 MB)
• The bonus problems are online: pdf (optional submission deadline for the exercises: July 29, 8:00).
• Concerning the oral exams:
  • As always: All of the material of this course is relevant for the exam (lectures, exercises, basic knowledge on Isabelle).
  • We covered a lot of material. Of course, I don't expect you to remember all the technical arguments at a ridiculous level of detail; however, you will need to know the key ideas and recurring proof patterns.
  • Concerning Isabelle: Clearly, during the exam there will not be enough time for live-coding in Isabelle. However, I might ask questions of the following type: How could one formalise XYZ in Isabelle? Which intermediate steps would be appropriate? What could be possible difficulties/pitfalls?
• Can't get enough of groups interacting with geometry, topology, and probability theory? In WS 20/21, there will be a seminar on L^2-invariants.
• Tired of measured equivalence relations? In WS 20/21, I will teach Differential Geometry I.
• The assignments for week 14 (the final week!) are online: pdf (submission deadline for the exercises: July 22, 8:00).
• Please register in FlexNow for the Studienleisung (and, if applicable also for the Prüfungsleistung).
• The results of the evaluation is available in GRIPS. Thanks for the feedback!
• In view of the COVID-19 pandemic, until further notice, this course will be taught remotely, based on
  • guided self-study (via extensive lecture notes),
  • interactive question sessions (via jitsi or zoom and the GRIPS forum),
  • remote exercise sessions (via jitsi or zoom and the GRIPS forum).
  More details: pdf.
• Currently, all non-virtual teaching at UR is suspended: more information about dealing with 2019-nCoV at UR; in particular, I cannot offer any in-person office hours or oral exams.

Ergodic Theory of Groups

Ergodic theory is the theory of dynamical systems, i.e., of measure preserving actions of groups on probability spaces. Such systems often occur in models of real-world phenomena. But also in theoretical mathematics, dynamical systems have a wide range of applications, e.g., in the following contexts:

• ubiquity of normal real numbers
• existence of arbitrarily long arithmetic sequences in sets of integers of positive density
• the computation of rank gradients of groups
• the computation of Betti number gradients of groups
• rigidity of lattices in Lie groups
• approximation properties of simplicial volume
• ...

In this course, we will introduce the basics of ergodic theory. We will then focus on group-theoretic properties and applications. Depending on the background and the interests of the audience, we might also discuss applications in geometric topology.

For additional excitement, we will aim at implementing a suitable fragment of the theory in a proof assistant (and thereby providing computer-verified proofs). Such tools are also used in the formalisation and verification of software systems.

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

Time/Location

Tuesday, 10:15--12:00, M 102,
Wednesday, 8:30--10:00, M 103.

In view of the COVID-19 pandemic, until further notice, this course will be taught remotely, based on:

• guided self-study (via extensive lecture notes),
• interactive question sessions (via jitsi or zoom and the GRIPS forum),
• remote exercise sessions (via jitsi or zoom and the GRIPS forum).

More details: pdf.

• The first "lecture" will be on Tuesday, April 21, 10:15. In this "lecture", we will get acquainted with the video conferencing tool, we will discuss organisational matters, and I will give a brief overview of this course.
• The last lecture is currently scheduled for Wednesday, July 22.
• Access information for the virtual meetings will be announced in GRIPS (and not publicly on the homepage).

Exercise class

Possibly: Friday, 10:15--12:00, M 104

• The exercise classes start in the second "first" week, on April 24; in this first session, some basics will be discussed.
• Details about the remote exercise sessions and the submission of solutions are explained in GRIPS.

Read me! Lecture notes, schedules and assignments

The lecture notes will grow during the semester and will be continuously updated. The first upload is scheduled for April 21, after the first lecture. In subsequent weeks with remote teaching, I will always try to provide the material for the whole week n + 1 on the Wednesday of week n.

• Lecture notes: pdf
• Correction log: here

Topics covered so far:

• Guide to the literature
• Introduction
  • What is ergodic theory?
  • Why ergodic theory?
  • Why proof assistants?
  • Overview of this course
• Dynamical systems
  • Dynamical systems
  • Standard examples
    [Rotations and shifts on the circle; Coset translations; Diagonal actions; Profinite completions; Bernoulli shifts]
  • Recurrence
    [Poincaré recurrence; Multiple recurrence; Application: Szemerédi's theorem]
  • Conjugacy and weak containment
    [Conjugacy; Factors and extensions; Weak containment]
  • Orbit equivalence and measure equivalence
    [Orbit equivalence; Measure equivalence]
• Ergodicity and mixing properties
  
  • Ergodicity and mixing properties
    [Ergodicity; Mixing actions; Invariant bounded functions; Invariant L^2-functions]
  • Ergodic theorems
    [Averaging a single transformation; The mean ergodic theorem; The pointwise ergodic theorem; Application: Decimal representations]
  • Ergodic decomposition
    [The space of invariant measures; The ergodic decomposition theorem; Sketch proof of ergodic decomposition]
• Rigidity: Cost
  
  • Measured equivalence relations
    [Standard equivalence relations; Two prototypical arguments; Measured equivalence relations; Ergodicity]
  • Cost
    [Graphings; Cost of measured equivalence relations; Basic cost estimates; Cost of free products; Application: Rigidity of free groups; Cost of products]
  • Cost of groups
    [Cost of groups; Weak containment and ergodic decomposition; The fixed price problem; The cost of the profinite completion; Application: Computation of rank gradients]
• Flexibility: Amenability
  
  • Amenable groups
    [Amenable groups: Invariant means; Amenable groups: Almost invariance; Amenable equivalence relations]
  • The dynamics of actions of \Z
    [Hyperfiniteness; The Rokhlin lemma; Dye's theorem]
  • The dynamics of actions of amenable groups
    [Hyperfiniteness; A dynamical characterisation of amenable groups; Application: Rank gradients of amenable groups; Ergodic theorems; The Rokhlin lemma]
• Integral approximation of simplicial volume
  
  • Simplicial volume
  • The residually finite view
  • The dynamical view
• Appendix
  
  • Measure theory
  • Amalgamated free products
  • A quick introduction to Isabelle
  • Some Isabelle fragments

This course will not follow a single book. Therefore, you should individually compose your own favourite selection of books. A list of suitable books can be found in the lecture notes.

Weekly assignments:

• Week 1 (announced on April 21): lectures of April 21/22, exercise series 0 (no submission; will be discussed in the exercise class on April 24)
• Week 2 (announced on April 22): lectures of April 28/29, exercise series 1 (submission before April 29, 8:00; will be discussed on "May 1")
• Week 3 (announced on April 29): lectures of May 05/06, exercise series 2 (submission before May 6, 8:00; will be discussed on May 8)
• Week 4 (announced on May 6): lectures of May 12/13, exercise series 3 (submission before May 13, 8:00; will be discussed on May 15)
• Week 5 (announced on May 13): lectures of May 19/20, exercise series 4 (submission before May 20, 8:00; will be discussed on May 22)
• Week 6 (announced on May 20): lectures of May 26/27, exercise series 5 (submission before May 27, 8:00; will be discussed on May 29)
• Week 7 (announced on May 27): lectures of June 3, exercise series 6 (submission before June 3, 8:00; will be discussed on June 5)
• Week 8 (announced on June 3): lectures of June 9/10, exercise series 7 (submission before June 10, 8:00; will be discussed on June 12)
• Week 9 (announced on June 10): lectures of June 16/17, exercise series 8 (submission before June 17, 8:00; will be discussed on June 19)
• Week 10 (announced on June 17): lectures of June 23/24, exercise series 9 (submission before June 24, 8:00; will be discussed on June 26)
• Week 11 (announced on June 24): lectures of June 30/July 1, exercise series 10 (submission before July 1, 8:00; will be discussed on July 3)
• Week 12 (announced on July 1): lectures of July 7/8, exercise series 11 (submission before July 8, 8:00; will be discussed on July 10)
• Week 13 (announced on July 8): lectures of July 14/15, exercise series 12 (submission before July 15, 8:00; will be discussed on July 17)
• Week 14 (announced on July 15): lectures of July 21/22, exercise series 13 (submission before July 22, 8:00; will be discussed on July 24)

Try me! Quick checks and interactive tools

• The quick checks in the lecture notes help to train elementary techniques and terminology. These problems should ideally be easy enough to be solved within a few minutes. Solutions are not to be submitted and will not be graded. These quickchecks will have feedback implemented directly in the pdf.
  This feature is based on PDF layers (not on JavaScript) and is supported by many PDF viewers, such as Acrobat Reader, Evince, Foxit Reader, Okular ("new" versions only), ...
• Moreover, there will be further interactive tools to explore dynamical systems.

Interactive tools:

• Digits.hs (experiments on distribution of digits; 21.04.2020)
• dynsys (exploration of rotation and shift actions on the circle)
• Super Blorx 3D (visualisation of products of rotation/shift actions on the 3-torus; requires a keyboard)
• P-Type (visualisation of 11-adic integers; requires a keyboard)
• P-Type II (visualisation of elements of Z_7 x Z_11; requires a keyboard)
• Blorx shift (visualisation of the standard Bernoulli shift; requires a keyboard)
• Ergodic averages (visualisation of ergodic averages for simple functions of rotation/shift actions on the circle)

Hack me! Implementation

We will implement a fragment of the theory in the proof assistant Isabelle.

• A toy example: Equivalence relations
  Version of the lecture on 10.06.: Equivalence relations
• Exercise 8.4: Orbits of equivalence relations
• Fragment: Graphs of functions
• Fragment: Generating sets of products
• Exercise 10.4: Ranks of products
• Fragment: Graphings
• Exercise 11.4: Graphings++
• Fragment: Graphings of orbit relations
• Fragment: Følner sequences
• Exercise 12.4: Følner sequences for finite groups
• Fragment: The standard odometer relation
• Exercise 13.4: A decomposition estimate

Ask me! Interactive sessions

There will be a forum and other communication tools linked on the GRIPS page of this course.

• All handwritten notes in a single file: pdf (22 MB)
• 21.04.2020: notes, Digits.hs
• 22.04.2020: notes
• 28.04.2020: notes
• 29.04.2020: notes
• 05.05.2020: notes
• 06.05.2020: notes
• 12.05.2020: notes
• 13.05.2020: notes
• 19.05.2020: notes
• 20.05.2020: notes
• 26.05.2020: notes
• 27.05.2020: notes
• 03.06.2020: notes
• 09.06.2020: notes
• 10.06.2020: notes, Equivalence_Relations.thy, Equivalence_Relations_Lecture.thy (lecture version)
• 16.06.2020: notes
• 17.06.2020: notes, RelRel_Iso.thy
• 23.06.2020: notes
• 24.06.2020: notes, Product_GenSet.thy
• 30.06.2020: notes
• 01.07.2020: notes, Graphing.thy
• 07.07.2020: notes
• 08.07.2020: notes, Graphing_OrbitRelation.thy
• 14.07.2020: notes
• 15.07.2020: notes, Folner.thy
• 21.07.2020: notes
• 22.07.2020: notes

Solve me! Exercise sheets

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

• Series 0, of April 21, 2020, no submission, will be discussed in the exercise classes on April 24
• Series 1, of April 22, 2020, submission before April 29, 8:00, will be discussed in the exercise classes on "May 1"
• Series 2, of April 29, 2020, submission before May 6, 8:00, will be discussed in the exercise classes on May 8
• Series 3, of May 6, 2020, submission before May 13, 8:00, will be discussed in the exercise classes on May 15
• Series 4, of May 13, 2020, submission before May 20, 8:00, will be discussed in the exercise classes on May 22
• Series 5, of May 20, 2020, submission before May 27, 8:00, will be discussed in the exercise classes on May 29
• Series 6, of May 27, 2020, submission before June 3, 8:00, will be discussed in the exercise classes on June 5
• Series 7, of June 3, 2020, submission before June 10, 8:00, will be discussed in the exercise classes on June 12
• Series 8, of June 10, 2020, submission before June 17, 8:00, will be discussed in the exercise classes on June 19
• Series 9, of June 17, 2020, submission before June 24, 8:00, will be discussed in the exercise classes on June 26
• Series 10, of June 24, 2020, submission before July 1, 8:00, will be discussed in the exercise classes on July 3
• Series 11, of July 1, 2020, submission before July 8, 8:00, will be discussed in the exercise classes on July 10
• Series 12, of July 8, 2020, submission before July 15, 8:00, will be discussed in the exercise classes on July 17
• Series 13, of July 15, 2020, submission before July 22, 8:00, will be discussed in the exercise classes on July 24
• Series 14, of July 22, 2020, optional submission before July 29, 8:00

Is it for me? 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, in basic group theory (as covered in the lectures on Algebra), and in probability theory (e.g., as in Wahrscheinlichkeitstheorie in SS 2012).
• Knowledge on algebraic topology (as in the course in WS 18/19) or group cohomology (as in the course in SS 19) is not necessary, but might allow us to treat more interesting applications.
• You could help me planning this course by filling in the (anonymous) questionnaire on background knowledge on the GRIPS page of this course: GRIPS

Last change: July 29, 2020.