Advanced Seminar on Group Theory

Lehrstuhl für Algebra und Zahlentheorie

Wintersemester 2021/22

Where & When:
The usual time is Tuesday 10:30 till 12:00 in seminar room 03.73.
Advanced topics in group theory are discussed in short series of talks and guest lectures.
Prof. Dr. Benjamin Klopsch & Martina Conte & Iker de las Heras & Moritz Petschick

The Congruence Subgroup Problem (programme)

The aim of this seminar is to give an elementary introduction to the congruence subgroup problem (CSP). We will mainly deal with the congruence subgroup problem for SLn and with some group-theoretic applications of it, following [4]. At the end we will survey some results and techniques concerning the congruence subgroup problem for more general linear algebraic groups. Other sources for learning about the subject are [1] and [3].
  1. Humphreys, J. E., Arithmetic Groups, Lecture Notes in Mathematics 789, Springer, Berlin, 1980
  2. Lubotzky, A., Group Presentations, p-Adic Analytic Groups and Lattices in SL2(C), Ann. of Math. 118, 1983
  3. Platonov V., Rapinchuk A.S., Algebraic groups and Number Theory, Pure and Applied Mathematics 139, Academic Press, San Diego, 1994
  4. Sury, B., The congruence subgroup problem. An elementary approach aimed at applications, Texts and Readings in Mathematics 24, Hindustan Book Agency, New Delhi, 2003
Tuesday 19.10.21
Seminar room 03.73
Martina Conte
The congruence subgroup problem for SL2(ℤ)
Tuesday 26.10.21
Seminar room 03.73
Moritz Petschick
The congruence subgroup problem for SL2(O)
Tuesday 02.11.21
Seminar room 03.73
Karthika Rajeev
The congruence subgroup problem for SLn(ℤ) with  n > 2.
Tuesday 09.11.21
Seminar room 03.73
Margherita Piccolo
The congruence subgroup problem for SLn(OS).
Tuesday 16.11.21
Seminar room 03.73
Iker de las Heras
The metaplectic kernel I.
Tuesday 23.11.21
Seminar room 03.73
Luis de Mendonça
The metaplectic kernel II.
Tuesday 30.11.21
Tuesday 07.12.21
No seminar!
Tuesday 14.12.21
online via webex
Marialaura Noce
(University of Göttingen)
Engel elements in groups of automorphisms of rooted trees
Groups of automorphisms of (d-adic) rooted trees have been studied for years as an important source of groups with interesting properties. For instance, the (first) Grigorchuk group is an example of a group where the set of left Engel elements is not a subgroup. In this talk we survey recent results about Engel conditions in some general families of groups of automorphisms of rooted trees. This is joint work with G.A. Fernandez-Alcober, A. Garetta, and G. Tracey.
Tuesday 21.12.21
No seminar!
Mon 10.01.22 14:30-16:00
Seminar room 02.81
Anitha Thillaisundaram
(University of Lund)
The Hausdorff dimensions of branch groups
The concept of Hausdorff dimension was defined in the 1930s and was originally applied to fractals and shapes in nature. However, from the work of Abercrombie, Barnea and Shalev in the 1990s, the computation of the Hausdorff dimensions in profinite groups has been made more doable. Starting with Abert and Virag's well-known result that there are groups acting on a rooted tree with all possible Hausdorff dimensions, mathematicians have been interested in computing the Hausdorff dimensions of explicit families of groups acting on rooted trees, and in particular, of the so-called branch groups. Branch groups first appeared in the context of the Burnside problem, where they delivered the first explicit examples of finitely generated infinite torsion groups. Since then, branch groups have gone on to play a key role in group theory and beyond. In this talk, we will survey known results concerning the Hausdorff dimensions of branch groups, in particular mentioning some recent joint work Gustavo Fernandez-Alcober and Sukran Gul.
Tuesday 18.01.22
Seminar room 03.73
Martina Conte, Luis de Mendonça, Moritz Petschick
Some applications of the Congruence Subgroup Property
Tuesday 25.01.22
Seminar room 03.73
Lilly Sandberger
Automorphismen von direkten Produkten endlicher Gruppen
Seminar room 03.73
Programme discussion for next semester
Tuesday 21.12.21
No seminar!

Right-angled Artin groups (programme)

The goal of this seminar is to give a gentle introduction to the theory of right-angled Artin groups, RAAGs for short. These groups span a wide range of groups from finitely generated free groups to finitely generated free abelian groups, and posses a rich structure of subgroups as well as nice algorithmic properties. The seminar will consist of five talks, each of them devoted to different topics exposed in the survey papers [4] and [11]; or in the book [8, Office hour 14].
  1. A. Baudisch, Kommutationsgleichungen in semifreien Gruppen, Acta Math. Hungarica 29(3-4) (1977), 235–249.
  2. A. Baudisch, Subgroups of semifree groups, Acta Math. Hungarica 81(1-4) (1979), 19–28.
  3. N. Bourbaki, Lie groups and Lie Algebras, Springer Berlin Heidelberg, 2002.
  4. R. Charney, An introduction to right-angled Artin groups, Geom. Dedicata 125(1) (2007), 141–158.
  5. J. Crisp, E. Godelle, B. Wiest, The conjugacy problem in subgroups of right-angled Artin groups, J. Topol. 2(3) (2009), 442–460.
  6. J. Crisp, L. Paris, The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group, Invent. math. 145 (2001), 19–36.
  7. M. W. Davis, T. Januszkiewicz, Right-angled Artin groups are commensurable with right-angled Coxeter groups, J. Pure Appl. Algebra 53 (2000), 229–235.
  8. A.J. Duncan, V.N. Remeslennikov, A.V. Treier, A survey of free partially commutative groups, J. Phys. Conf. Ser. 1441 (2020), 012136.
  9. C. Droms, Subgroups of graph groups, J. Algebra 110(2) (1987), 519–522.
  10. C. Droms, Graph Groups, Coherence, and Three-Manifolds, J. Algebra 106 (1987), 484–489.
  11. D. Margalit, M. Clay, Office Hours with a Geometric Group Theorist, Princeton University Press, 2017.
  12. H. Servatius, Automorphisms of graph groups, J. Algebra 126 (1989), 34–60.
Wednesday 23.02.22
Seminar room 03.73
Iker de las Heras
The word and conjugacy problems in RAAGs
Wednesday 02.03.22
Seminar room 03.73
Moritz Petschick
Centralizers of RAAGs
Wednesday 09.03.22
Seminar room 03.73
Margherita Piccolo
Special subgroups of RAAGs
Wed 16.03.22 14:30-16:00
Seminar room 03.73
Lucas Ruhstorfer
Local-global conjectures
Monday 21.03.22
Seminar room 03.73
Martina Conte
Tits’ conjecture