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Advanced topics in Geometric Group Theory (Summer Semester 2024)


Premeeting: Tuesday 20.02.2024, 14:00-15:00 Uhr, Room: SR 2.058

Seminar dates: Blockseminar after the end of term (in the first half of August 2024, tentatively in the week 05.08. - 09.08.)

Language: Talks can be given in English or German

Prerequisites: Knowledge of the course "Geometric Group Theory" will be assumed. For some talks some knowledge of the course "Geometric Group Theory II" may be helpful. Lecture notes for both courses are available on the ILIAS course webpages.

Registration: To register or if you have any questions, please write to claudio.llosa@kit.edu, before the premeeting.


Geometric group theory studies the interactions between finitely generated groups and geometric spaces, creating a connection between algebra and geometry. We will explore advanced topics in geometric group theory, building on notions and ideas introduced in the lecture course. Broadly we will look at two types of topics:

  • Explicit examples: Explicit examples play an important role in geometric group theory, as they can reveal interesting phenomena and pathologies. Such examples include groups that arise from (quasi) actions on trees like Grigorchuk's group, an infinite torsion group of intermediate growth, and Thompson's group F, a group of exponential growth that is not solvable, does not contain any free subgroups and for which it is famously not known whether it is amenable. Another class of groups that we may study closer are right-angled Artin groups, which are a class of {\rm CAT}(0) groups that have played a central role in recent developments in low-dimensional topology, including Agol's proof of Thurston's virtual fibring conjecture.
  • Quasi-isometry invariants: Aside from the possibility to deepen our understanding of quasi-isometry invariants that we have already encountered, such as hyperbolicity, we will also be able to explore new invariants. New examples may include semihyperbolicity and Dehn functions. Semihyperbolicity provides a natural generalisation of being {\rm CAT}(0) in a similar way that hyperbolicity provides a generalisation of being {\rm CAT}(-1). Dehn functions on the other hand provide a fine quasi-isometry invariant that is a quantitative measure for the complexity of the word problem in finitely presented groups.

Format: We will cover a selection of the aforementioned examples and invariants. We will determine the precise choice together in the premeeting, where topics will be assigned. The topics are modular and some of them can be explored individually, while others are suitable for exploration in small groups. Where topics are explored in small groups, every group member will give an individual 60 minute talk on an aspect of the topic.


To get an idea of the possible topics you may consult the following literature. More precise information on topics of talks and corresponding literature will be provided to participants before the premeeting. Some individual talks may require further literature that will be provided by us.

  • J.M. Alonso, M.R. Bridson, Semihyperbolic groups, Proc. London Math. Soc. (3)70(1995), no.1, 56–114.
  • 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.
  • J.W. Cannon, W.J. Floyd, W.R. Parry, Introductory notes on Richard Thompson's groups, Enseign. Math. (2) 42 (1996), no. 3-4, 215-256.
  • Office hours with a geometric group theorist, edited by Matt Clay and Dan Margalit, Princeton University Press, Princeton, NJ, 2017.
  • P. de la Harpe, Topics in Geometric Group Theory, Chicago Lectures in Mathematics, University of Chicago Press, 2003.
  • T. Koberda, Right-angled Artin groups and their subgroups, https://users.math.yale.edu/users/koberda/raagcourse.pdf.
  • J. Meier, Groups, Graphs and Trees - An Introduction to the Geometry of Infinite Groups, London Mathematical Society Student Texts 73, Cambridge University Press, 2008.