### Geometric Group Theory (Summer Semester 2023)

• Lecturer:
• 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.

Further information will be available on the ILIAS page of the course soon. You will also find the problem sheets and other course material there.

Schedule
Lecture: Thursday 8:00-9:30 20.30 -01.12
Friday 9:45-11:15 20.30 SR 3.68
Problem class: Monday 11:30-13:00 20.30 SR 3.68
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 Dr. Gabriel Pallier
Office hours: On appointment.
Room 1.015 Kollegiengebäude Mathematik (20.30)
Email: gabriel.pallier@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 be assumed. Besides this the module "Einführung in die Algebra und Zahlentheorie" is helpful, in particular, this concerns the familiarity with basic concepts in group theory.

# References

• C. Löh, "Geometric Group Theory - An Introduction", Universitext, Springer Verlag, 2017.
• A. Hatcher "Algebraic Topology", Cambridge University Press, 2006.
• 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.