Advanced Seminar on Group Theory

Lehrstuhl für Algebra und Zahlentheorie

Sommer semester 2026

Where & When:
The usual time is Tuesday 10:30 - 12:00 in seminar room 03.73.
Content:
Advanced topics in group theory are discussed in a series of talks and guest lectures.
Organizers:
Prof. Dr. Benjamin Klopsch & Alena Ramona Meyer & Dr. Mikel Eguzki Garciarena

Hyperbolic Groups

The main idea of geometric group theory is that groups can be thought of and studied as geometric objects. One way to do this is by introducing a metric structure on groups via word metrics on their Cayley graphs, which then allows us to study the large scale geometry of groups with respect to this metric structure.

In particular, one can introduce a notion of negative curvature to large scale geometry via slim triangles, which can be applied to Cayley graphs. This then leads to the definition of hyperbolic groups as those finitely generated groups with negative curvature.

The study of hyperbolic groups will be the main focus of our Seminar. In the first half (Talks 1-4) we will introduce all the basic notions required to define hyperbolic groups. In the second half of the seminar (Talks 5-9) we will look at some interesting properties and applications, including the Rips construction and the solvability of the word problem for hyperbolic groups.

More details can be found in the program.

References:

  1. C. Löh. Geometric group theory. An introduction, Universitext. Springer, 2017. Springer
  2. R. Lyndon, P. Schupp. Combinatorial Group Theory, Springer, Berlin, 1977. Springer
  3. E. Ghys, P. de la Harpe (eds.). Sur les groupes hyperboliques d’après Mikhael Gromov, Birkhäuser, 1990. Springer
  4. M. Hull. Hyperbolic Groups, Lecture Notes. PDF
  5. E. Rips. Subgroups of small cancellation groups, Bull. London Math. Soc. 14 (1982), 45–47. DOI
Tue 14.04.26
Seminar room 03.73
Holger Kammeyer
Profiniteness of higher rank volume
While many properties such as amenability, property (T) and FA, finiteness properties, Euler characteristic,... can differ for profinitely isomorphic groups, we present a result in the positive direction: The covolume of lattices in higher rank Lie groups with the congruence subgroup property is determined by the profinite completion. Without relying on CSP, we additionally show that volume is a profinite invariant of octonionic hyperbolic congruence manifolds. Joint work with Steffen Kionke and Ralf Köhl.
Tue 21.04.26
Seminar room 03.73
Hanna Hecker
Cayley graphs and (quasi-)isometries.
Tue 28.04.26
Seminar room 03.73
Daniel Morlock
Quasi-geodesics and quasi-isometry invariants.
Tue 05.05.26
Seminar room 03.73
Philipp Hauchwitz
(Quasi-)Hyperbolic spaces.
Tue 12.05.26
Seminar room 03.73
Moritz Lummerich
Hyperbolic groups.
Tue 19.05.26
Seminar room 03.73
Chany Genua
The word problem in hyperbolic groups.
Tue 26.05.26
Seminar room 03.73
Sofiya Yatsyna
(Royal Holloway,
University of London)
Cohomological Finiteness Properties of Almost Automorphism Groups

In 1965, Thompson introduced a group V, with subgroups F ≤ T ≤ V, to provide (counter-)examples of finitely presented groups for certain conjectures. Today, they form a very important class of groups with many interesting and uncommon properties. One such property, shown by Brown and Geoghegan in 1982, shows that F is the first known group of type FP∞ whose cohomology vanishes with coefficients in the group ring ℤF.

In this talk, we will focus on the totally disconnected locally compact analogue of Thompson's F, known as almost automorphism groups, and building from the work of Sauer and Thurmann, we will show that these almost automorphism groups satisfy similar cohomological properties. This is joint work with Laura Bonn, Bianca Marchionna and Lewis Molyneux.

Tue 02.06.26
Seminar room 03.73
Jan Ziske
Elements of infinite order.
Tue 09.06.26
No seminar
No seminar.
Tue 16.06.26
Seminar room 03.73
Hendrik Untch
Centralisers in hyperbolic groups.
Tue 23.06.26
Seminar room 03.73
Clotilde Gauthier
Small cancellation theory.
Tue 30.06.26
Seminar room 03.73
Daniel Echtler
The Rips Construction.
Tue 07.07.26
Seminar room 03.73
Nadja Valentin
Radical in Groups with a descending chain condition on centralizers

Groups in which there is no infinite strictly descending chain of centralizer, so called Mc-groups, have been studied by Bryant and Hartley in the late 70's. Of special interest in this context are nilpotent and solvable subgroups and their corresponding radicals: The Fitting subgroup (the group generated by all normal nilpotent subgroups) and the solvable radical (the group generated by all normal solvable subgroups). Note that these are not necessarily nilpotent or solvable respectively. In 1997 Derakshan and Wagner proved, using tools from Model Theory, that the Fitting subgroup of Mc-groups are nilpotent. More recently in 2009, Khrukro showed that in case of pseudofinite Mc-groups, the solvable radical is solvable.
We are interested in relaxing the constraints and moving to groups where there is no infinite descending chain of centralizer each having infinite index in their predecessors.

In this talk we will give an overview on the results known for the existence or non-existence of the above mentioned radicals in groups with descending chains on definable subgroups and their connection to tameness conditions in Model Theory.

Tue 14.07.26
Seminar room 03.73
TBA
Guest Talk.
Tue 21.07.26
Seminar room 03.73
TBA
Guest Talk.

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