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Seminar on Geometric Group Theory (Sommersemester 2022)

Seminar: Blockveranstaltung Mathematik
Seminarleitung Jun.-Prof. Dr. Claudio Llosa Isenrich
Sprechstunde: nach Vereinbarung
Zimmer 1.005 Kollegiengebäude Mathematik (20.30)
Email: claudio.llosa@kit.edu
Seminarleitung Dr. Julia Heller
Sprechstunde: nach Vereinbarung oder spontan
Zimmer 1.008 Kollegiengebäude Mathematik (20.30)
Email: julia.heller@kit.edu

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.

Filling of a loop in a Cayley graph

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.

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 from 14:00 - 15:00 via Zoom.

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.

Talks can be given in English or German.


  • 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.
  • T.W. Gamelin, R.E. Greene, "Introduction to Topology", 2nd Edition, Dover, 1999.
  • A. Hatcher, "Algebraic Topology", 2002.
  • C. Löh, "Geometric Group Theory - An Introduction", Universitext, Springer Verlag, 2017.
  • H. Short (editor), "Notes on word hyperbolic groups", Group Theory from a Geometric Viewpoint, Proc. ICTP Trieste 1990.
  • J. Meier, "Groups, graphs and trees", London Mathematical Society Student Texts vol. 73, Cambridge University Press, 2008.