# 401-4531-66L Topics in Rigidity Theory

Semester | Autumn Semester 2020 |

Lecturers | M. Burger |

Periodicity | non-recurring course |

Language of instruction | English |

### Courses

Number | Title | Hours | Lecturers | |||||||
---|---|---|---|---|---|---|---|---|---|---|

401-4531-66 V | 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. | 3 hrs |
| M. Burger |

### Catalogue data

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. |

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. |

Prerequisites / Notice | For this course some knowledge of elementary Lie theory would be good. We will however treat Lie groups by examples and avoid structure theory since this is not the point of the course nor of the techniques. |

### Performance assessment

Performance assessment information (valid until the course unit is held again) | |

Performance assessment as a semester course | |

ECTS credits | 6 credits |

Examiners | M. Burger |

Type | session examination |

Language of examination | English |

Repetition | The performance assessment is offered every session. Repetition possible without re-enrolling for the course unit. |

Mode of examination | oral 20 minutes |

This information can be updated until the beginning of the semester; information on the examination timetable is binding. |

### Learning materials

Main link | Information |

Only public learning materials are listed. |

### Groups

No information on groups available. |

### Restrictions

There are no additional restrictions for the registration. |

### Offered in

Programme | Section | Type | |
---|---|---|---|

Doctoral Department of Mathematics | Graduate School | W | |

Mathematics Master | Selection: Geometry | W |