### Geometric Group Theory (Summer Semester 2021)

- Lecturer: Jun.-Prof. Dr. Claudio Llosa Isenrich
- Classes: Lecture (0153300), Problem class (0153310)
- Weekly hours: 4+2

This course will be offered in English. You find a brief description of the contents and prerequisites, as well as some literature at the end of this page.

The lectures and problem classes will take place online via zoom. The zoom links, problem sheets and further information will be posted on the ILIAS page of the course.

Please contact us via email if you encounter any technical problems in accessing the ILIAS page.

Schedule | |||
---|---|---|---|

Lecture: | Monday 12:00 - 13:30 | Online via Zoom | Begin: 12.4.2021, End: 20.7.2021 |

Tuesday 10:00 - 11:30 | Online via Zoom | ||

Problem class: | Friday 12:00 - 13:30 | Online via Zoom | Begin: 16.4.2021, End: 23.7.2021 |

Lecturers | ||
---|---|---|

Lecturer | Jun.-Prof. Dr. Claudio Llosa Isenrich | |

Office hours: by appointment | ||

Room 1.005 Kollegiengebäude Mathematik (20.30) | ||

Email: claudio.llosa@kit.edu | ||

Problem classes | M. Sc. Jerónimo García Mejía | |

Office hours: by appointment | ||

Room 1.012 Kollegiengebäude Mathematik (20.30) | ||

Email: jeronimo.garcia@kit.edu |

**Contents**

This course will provide an introduction to geometric group theory, which studies the interactions between finitely generated groups and geometric spaces, creating connections between algebra and geometry. While a priori groups may seem like purely algebraic objects, they can naturally arise as symmetries of geometric objects. For instance, the symmetries of a regular n-gon form a group (the dihedral group ). In fact, every finitely generated group admits a natural action by isometries on a metric space, known as its Cayley graph. For instance the Cayley graph of the integers is the real line with vertices given by the integer points and the group action defined by translation.

Studying group actions on geometric spaces, allows us to gain insights into "the geometry of groups". Conversely, knowing that a geometric space admits an interesting group action allows us to obtain a better understanding of the space itself. Over the last decades, these interactions between group theory and geometry have led to an array of fundamental results in both areas. This course will provide an introduction to these interactions and their consequences.

In particular, we will learn about:

- finitely generated groups and group presentations
- Cayley graphs and group actions
- quasi-isometries of metric spaces, quasi-isometry invariants and the Theorem of Schwarz-Milnor
- explicit examples of infinite groups and their connections to geometry

**Prerequisites**

Knowledge of the contents of the module "Elementare Geometrie" or "Einführung in die Topologie und Geometrie" is recommended. In particular, knowledge of the basic concepts on metric and topological spaces will is recommended. Besides this the module "Einführung in die Algebra und Zahlentheorie" is helpful. In particular, some familiarity with the basic concepts in group theory will be very helpful.

# Examination

The exam format will be oral. Further details will be announced in the lecture, closer to the end of term.

# References

- C. Löh,
*"Geometric Group Theory - An Introduction"*, Universitext, Springer Verlag, 2017. - J.-P. Serre
*"Trees"*, Springer Monographs in Mathematics, Springer Verlag, 1980. - J. Meier,
*"Groups, graphs and trees"*, London Mathematical Society Student Texts vol. 73, Cambridge University Press, 2008. *"Office hours with a geometric group theorist"*, edited by Matt Clay and Dan Margalit, Princeton University Press, Princeton, NJ, 2017.- M.R. Bridson, A. Haefliger,
*"Metric spaces of non-positive curvature"*, Grundlehren der mathematischen Wissenschaften vol. 319, Springer Verlag, 1999. - C. Drutu, M. Kapovich, with an Appendix by B. Nica,
*"Geometric Group Theory"*, AMS Colloquium publications vol.63, 2018.