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401-4531-66L 6 Credits DR , MSC D-MATH

Topics in Rigidity Theory

Lecturers & Examiners: Prof. em. Dr. Marc Burger
VVZ CR n/a

Last Updated: 2026-02-05 15:34:47

Abstract

The aim of this course is to give detailed proofs of Margulis' normal subgroup theorem and his superrigidity theorem for lattices in higher rank Lie groups.

Objective

Understand the basic techniques of rigidity theory.

Content

This course gives an introduction to rigidity theory, which is a set of techniques initially invented to understand the structure of a certain class of discrete subgroups of Lie groups, called lattices, and currently used in more general contexts of groups arising as isometries of non-positively curved geometries. A prominent example of a lattice in the Lie group SL(n, R) is the group SL(n, Z) of integer n x n matrices with determinant 1. Prominent questions concerning this group are: - Describe all its proper quotients. - Classify all its finite dimensional linear representations. - More generally, can this group act by diffeomorphisms on "small" manifolds like the circle? - Does its Cayley graph considered as a metric space at large scale contain enough information to recover the group structure? In this course we will give detailed treatment for the answers to the first two questions; they are respectively Margulis' normal subgroup theorem and Margulis' superrigidity theorem. These results, valid for all lattices in simple Lie groups of rank at least 2 --like SL(n, R), with n at least 3-- lead to the arithmeticity theorem, which says that all lattices are obtained by an arithmetic construction.

Resources

Literature

- R. Zimmer: "Ergodic Theory and Semisimple groups", Birkhauser 1984. - D. Witte-Morris: "Introduction to Arithmetic groups", available on Arxiv - Y. Benoist: "Five lectures on lattices in semisimple Lie groups", available on his homepage. - M.Burger: "Rigidity and Arithmeticity", European School of Group Theory, 1996, handwritten notes, will be put online.

Learning Materials (Links)

General Information

Language
English
Levels
DR , MSC

Examination

Type
session examination
Mode
oral 20 minutes

Course Components

Type Title Time & Place Hours
lecture Topics in Rigidity Theory
"Hybrid" At most 29 students may attend the class in HG G 19.1, further students would have to follow online. Online for all students as of November 2020.
  • Thu 14:15-17:00 (HG G 19.1)
  • 17.09 Date 14:15-17:00 (HG F 3)
3 h weekly

Offered In

  • Mathematics Master
    • Electives (For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.)
  • Doctoral Department of Mathematics (More Information at: The list of courses (together with the allocated credit points) eligible for doctoral students is published each semester in the newsletter of the ZGSM. WARNING: Do not mistake ECTS credits for credit points for doctoral studies!)