## Lecture## Lokally Compact Groups## Winter term 2024/25## Prof.Dr. Linus Kramer## with Raquel Murat García |

**Course Locally Compact Groups**

Locally compact groups are topological groups, i.e. groups which carry a topology such that multiplication and inversion are continuous maps. They play a role in different areas of mathematics such as geometry, geometric group theory, Lie theory, operator theory or harmonic analysis. We first study general topics in topological groups such as subgroups, quotients, connectedness, and actions. Then we consider profinite groups and van Dantzig's theorem. After this we turn to the more advanced structure theory. We will introduce the Haar integral and use it to prove the Peter-Weyl theorem about the structure of compact groups. In the last part of the course we will consider Pontrjagin-Van Kampen duality of compact and locally compact abelian groups.

**Audience:** The course is aimed at advanced BSC students and MSC students.
The topic is well suitable for a master's thesis (or a bachlor's thesis).

**Prerequisites:** Basic algebra (groups, rings, vector spaces and modules)
and a solid background in point-set topology (as covered in the course
*Grundlagen der Analysis, Topologie und Geometrie*).
Knowledge about Lie groups or functional analysis is certainly helpful, but not required
for this course.

The **class** takes place on Tuesday and Friday 8:15 - 10:00 Uhr in lecture room M4.
**We begin on Tuesday Di 8.10.2024 at 8:15**.
We have set up a course in the Learnweb for the class and the tutorial.
The Learnweb key will be anounced in the class.

There will be a weekly **tutorial** for the class. It is essential
that you participate actively in the tutorial and that you do the homework problems.
The date for the tutorial will be fixed during the first week of classes.

#### Literature

- Außenhofer, Dikranjan, Giordano Bruno, Topological groups and the Pontryagin-van Kampen duality
- Deitmar, Echterhoff, Principles of harmonic analysis
- Dugundji, Topology
- Hewitt, Ross, Abstract harmonic analysis I
- Hofmann, Morris, The structure of compact groups
- Hofmann, Morris, The Lie theory of connected pro-Lie groups
- Stroppel, Locally compact groups
- Wilson, Profinite groups

#### Exercise sheets

Exercise sheet 01 |

Exercise sheet 02 |

Exercise sheet 03 |

Exercise sheet 04 |

Exercise sheet 05 |

Exercise sheet 06 |

#### Course notes

Frontmatter |

Chapter 1 |

Chapter 2 |

There is also a book manuscript in preparation..