# Course Identification

Analytic group theory
20214201

## Lecturers and Teaching Assistants

Prof. Tsachik Gelander
Omer Lavi

## Course Schedule and Location

2021
First Semester
Wednesday, 10:15 - 13:00
28/10/2020

## Field of Study, Course Type and Credit Points

Mathematics and Computer Science: Lecture; Elective; Regular; 3.00 points

N/A

## Prerequisites

1. Basic Group Theory.

2. Topology.

3. Measure Theory.

20

English

## Attendance and participation

Obligatory

Numerical (out of 100)

50%
25%
25%

Final assignment

N/A
N/A
-
N/A

N/A

## Syllabus

I Amenability

1 The Hausdorff Banach Tarski Paradox

1.1 Equidecomposability

1.3 The Sphere

1.4 Summary

1.5 More on F2 inside SO (3)

2 The Ping-Pong Lemma

2.1 First Formulation

2.2 Second Formulation

3 Amenability

3.1 Definition

3.2 Means

3.3 Banach Spaces

3.4 Asymptotically Invariant Nets

3.5 Følner Nets

3.6 Cayley Graphs

3.7 Back to Amenability

3.8 The Følner Condition

3.9 Abelian Groups

3.10 Summary

4 Application of Amenability

4.1 Banach Limits

4.2 Von Neumann Ergodic Theorem

5 Some Group Theory

5.1 Growth

5.2 Nilpotent Groups

6 Actions of Amenable Groups

6.1 Affine Actions

6.2 Actions on Compact Hausdorff Spaces

7 Elementary Amenably Groups

7.1 The Class of Elementary Amenable Groups

7.2 Subgroups

7.3 Quotient Groups

7.4 Extensions

7.5 Direct Limits

8 Topological Groups

8.1 Definition

8.2 Haar Measure

8.3 Amenability of Topological Groups

II Kazhdan’s Property (T)

1 A Bit of History

2 Property (T)

2.1 Definition and First Results

2.2 Induced Representation

2.3 Proof of Theorem II.2.1.13

2.4 Expanders

3 Property (FH)

3.1 Isometric Actions

3.2 Property (FH)

3.3 Back to Property (T)

3.4 Hyperbolic Space

4 SL(2,Qp) and its tree

4.2 The tree of SL(2,Qp)

5 Actions on Manifolds

5.1 The Jordan Theorem

5.2 Bierberbach Theorem

5.3 The Kazhdan-Margulis Theorem

## Learning Outcomes

Analytic group theory is an active area which is strongly related to many fields of mathematics. It has applications in Analysis, Topology, Geometry, Combinatorics, Computer Science, Number theory and other topics.