## Arithmetic groups

### Practical information

Semester: 2017/2018 - summerTimes:

Tuesdays 10:00 - 12:00, Seminarraum 0.003, Endenicher Allee 60

Thursdays 10:00 - 12:00, Seminarraum 0.008, Endenicher Allee 60.

**NB: The lectures for this course have finished.**

If you have any questions, either administrative or mathematical, do not hesitate to contact me by email.

### Exams

The exam will be oral and will be held in office 2.003 at the Endenicher Allee 60.If you have not yet picked a time for your exam, please send me an email.

### Contents

Arithmetic groups are groups that are obtained as the integer points of an algebraic group (an example is the group SL(2,Z)). In this course we will study arithmetic subgroups of linear Lie groups. These groups give rise to many interesting examples of locally symmetric spaces.The following is a preliminary list of topics we will cover:

- Definition, properties and examples

- Arithmetic groups in hyperbolic geometry

- Construction of non-arithmetic lattices in SO(n,1)

- Higher rank lattices

- Arithmetic groups coming from hypergeometric differential equations.

### Preliminaries

Necessary: Basic (linear) algebra and analysis, basic differential geometryUseful: some knowledge of Lie groups.

### Material

We will mainly be following the book*Introduction to Arithmetic Groups*by Dave Witte Morris, which is freely available here.

All Section numbers below refer to [1], unless otherwise specified.

Lecture | Sections |

10 / 04 | 1.1, 1.3 |

12 / 04 | 1.1, 1.2 |

17 / 04 | 1.2, 1.3 |

19 / 04 | A.3, 4.1, 4.2 |

24 / 04 | A.2, 4.2, 4.3, 4.4 |

03 / 05 | 4.4, 4.5, 4.6 |

08 / 05 | 4.5, 4.6, A.4 |

15 / 05 | 4.7, 4.8, 5.1 |

17 / 05 | 5.2, 5.3 |

29 / 05 | 5.4, 5.5 |

05 / 06 | 5.5 |

07 / 06 | 5.5, 6.1 |

12 / 06 | 6.2 |

14 / 06 | Chapter 7 of [2], pages 37, 38 of [3] |

19 / 06 | Pages 44, 45 of [3] |

21 / 06 | 6.4, 6.5 |

26 / 06 | 6.5 |

28 / 06 | 6.5, 6.6 |

03 / 07 | 7.1, 7.2 |

05 / 07 | 7.3, 8.1, 16.1, 16.2 |

10 / 07 | 16.3, 16.5 |

**References**

[1] | Dave Witte Morris, Introduction to Arithmetic Groups. Deductive Press, 2015. |

[2] | A. F. Beardon, The geometry of discrete groups. Springer, 1983. |

[3] | P. Buser and P. Sarnak, On the period matrix of a Riemann surface of large genus. Inventiones mathematicae, 117: 27 - 56, 1994. |