### Seminar on Geometric Group Theory (Summer Semester 2022)

- Lecturer: Jun.-Prof. Dr. Claudio Llosa Isenrich
- Classes: Seminar (0172700)
- Weekly hours: 2

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

Seminar: | en-block class | Mathematik |

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

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

Office hours: by appointment | ||

Room 1.005 Kollegiengebäude Mathematik (20.30) | ||

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

Lecturer | Dr. Julia Heller | |

Office hours: | ||

Room 1.008 Kollegiengebäude Mathematik (20.30) | ||

Email: julia.heller@kit.edu |

**Contents**

Geometric Group Theory studies the interactions between finitely generated groups and geometric spaces, creating a connection between Algebra and Geometry. Such interactions arise when groups act nicely on geometric spaces. Every finitely generated group admits such a "nice" action on a graph, called its Cayley graph. This allows us to equip every group with a metric, which we call its word metric. Roughly speaking, a geometric property of a group is a property of this metric space.

The goal of this seminar will be to explore this connection between Geometry and Algebra by studying geometric invariants of groups. The aim is to complement a first lecture course on Geometric Group Theory by introducing concepts that were not discussed there. In doing so we plan to focus on non-positively curved groups, but not limit ourselves to them. Invariants that may be considered include Dehn functions, boundaries, ends, and amenability.

**Prerequisites**

Knowledge of the contents of the course "Elementare Geometrie", as well as some basic notions in Geometric Group Theory, such as finite presentations, Cayley graphs and quasi-isometries will be useful.

For motivated students who have not attended a course in Geometric Group theory we will offer a short introduction (ca. 3 lectures) during the semester, in which we will summarise without proofs some key concepts and results and provide some examples.

**Registration**

If you are interested in participating in this seminar, please contact Claudio Llosa (claudio.llosa@kit.edu) before the preliminary meeting.

**Dates and Times**

The preliminary meeting (Vorbesprechung) will take place on Tuesday, 15.02.2022, at 14:00.

The seminar is planned as a block seminar at the beginning of the break (probably in the week 01.08.-05.08.). There is some flexibility in the precise dates and times and we will try to adjust them to the participants availabilities, where this is possible.

**Language**

Talks can be given in English or German.

# References

- N. Brady, T.Riley, H. Short,
*"The geometry of the word problem for finitely generated groups"*, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2007. - M.R. Bridson, A. Haefliger,
*"Metric spaces of non-positive curvature"*, Grundlehren der mathematischen Wissenschaften vol. 319, Springer Verlag, 1999. - M.R. Bridson,
*"The geometry of the word problem"*, Course notes, Invitations to geometry and topology, 29–91, Oxf. Grad. Texts Math., 7, Oxford Univ. Press, Oxford, 2002. *"Office hours with a geometric group theorist"*, edited by Matt Clay and Dan Margalit, Princeton University Press, Princeton, NJ, 2017.- C. Löh,
*"Geometric Group Theory - An Introduction"*, Universitext, Springer Verlag, 2017. - J. Meier,
*"Groups, graphs and trees"*, London Mathematical Society Student Texts vol. 73, Cambridge University Press, 2008.