Advanced Seminar on Group Theory

Lehrstuhl für Algebra und Zahlentheorie

Wintersemester 2024/25

Where & When:
The usual time is Tuesday 10:30 - 12:00 in seminar room 03.73.
Content:
Advanced topics in and around group theory are discussed in form of a reading course.
Organizers:
Prof. Dr. Benjamin Klopsch & Doris Grothusmann

Tue 15.10.24
Seminar room 03.73
all of us
Discussion of topics for this semester

Braid groups

Braid groups are well known for being studied in many different areas of Mathe- matics thanks to the variety of approaches one can use to define them.

During the seminar we see most of these approaches, and we learn how they were used through the years to prove relevant properties such as torsion-freeness, hopfi- anity and residual finiteness and furthermore, to answer the word and conjugacy problems.

For our purposes, we follow the notes by Gonzalez-Meneses [1], which cover almost all the basics and collect often multiple proofs of the same results, so to gain a parallel view over all the approaches.

Due to the conciseness of such notes, I suggest to integrate the material -whenever required- with Chapter 1 of Kassel-Turaev’s textbook [2] and/or follow the references already cited in [1].

See the programme for more details.

Main reference:

  1. J. Gonzalez-Meneses, Basic results on braid groups, 2011. link
  2. C. Kassel and V. Turaev, Braid groups, 2024. link
Tue 22.10.24
Seminar room 03.73
Giada Serafini
Braids and braid diagrams. Braid groups, configuration spaces and mapping class groups.
Tue 29.10.24
Seminar room 03.73
Ilaria Castellano
Braids as automorphisms of the free group. First solution to the word problem. Residual finiteness and Hopfianity.
Tue 05.11.24
Seminar room 03.73
Doris Grothusmann
Braid groups are torsion free.
Tue 12.11.24
Seminar room 03.73
Margherita Piccolo
Left orderability of braid groups. Presentations of braid groups.
Tue 19.11.24
Seminar room 03.73
tba
Garside structures. Center of braid groups. Conjugacy problem.
Tue 26.11.24
Seminar room 03.73
tba
Braid groups (6)
Tue 03.12.24
Seminar room 03.73
tba
tba
Tue 10.12.24
Seminar room 03.73
tba
tba
Tue 17.12.24
Seminar room 03.73
tba
tba

The Sigma-invariants

The idea is to have an introduction to the classical theory of Sigma-invariants for discrete groups, with a particular focus on the first invariant Σ1, mentioning some of the applications in group theory. If time will permit, we will conclude with a talk on the generalisation of Σ1 to locally compact groups.

Tue 07.01.25
Seminar room 03.73
tba
The Sigma-invariants (1)
Tue 14.01.25
Seminar room 03.73
tba
The Sigma-invariants (2)
Tue 21.01.25
Seminar room 03.73
tba
The Sigma-invariants (3)
Tue 28.01.25
Seminar room 03.73
tba
The Sigma-invariants (4)

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