# Advanced Seminar on Group Theory

Lehrstuhl für Algebra und Zahlentheorie

## Wintersemester 2020/2021

Where & When:
The usual time is Tuesday 10:30 – 11:30, for the time being online.
Content:
Advanced topics in group theory are discussed in short series of talks and guest lectures.

## Verbal Width in Groups (programme)

The width m(w, G) of a word w in a group G is the maximal distance to 1 of any element of the Cayley graph of the verbal subgroup w(G) with respect to the natural generating set of all w-values Gw. An open problem is to characterise the words w where for all finite groups G the growth of m(w, G) is bounded in terms of the rank of G. Our goal is to understand the solution of this problem for finite nilpotent groups, as given in the exposition .

1. Segal, Dan. Words: notes on verbal width in groups. London Mathematical Society Lecture Note Series, 361. Cambridge University Press, Cambridge, 2009. Available at the library.
Tuesday 03.11.20
Lecture Hall 5L
Dominic Witt
Generalities and width of lower central words in nilpotent groups
Tuesday 10.11.20
Lecture Hall 5L
Moritz Petschick
Words of infinite width
Tuesday 17.11.20
Lecture Hall 5L
Margherita Piccolo
Finitely generated virtually abelian-by-nilpotent groups are verbally elliptic
Tuesday 17.11.20, 14:30
Lecture Hall 5H
Luca Di Gravina
The Möbius function of subgroup lattices I
The Möbius function and the related inversion formula are classical tools in number theory. If we regard the set of natural numbers as a poset ordered by divisibility, we realize that the Möbius function can be generalized to sets that are partially ordered and locally finite, as ℕ is. The study of the properties of such a function is of great interest in combinatorics and it relies on the characterization of order structures. I will present this generalization for the subgroup lattice of a finitely generated profinite group and a related conjecture for those which are positively finitely generated (i.e., the probability that some group elements generate the whole group is > 0). This conjecture can be reduced to a problem that only regards finite almost-simple groups. Part of my research involves the study of such a reduction in the case of finite classical groups. In particular, I will introduce some methods and ideas to be applied to PSL(n,q) and PGL(n,q).
Monday 23.11.20, 14:30
Lecture Hall 5H
Luca Di Gravina
The Möbius function of subgroup lattices II
The Möbius function and the related inversion formula are classical tools in number theory. If we regard the set of natural numbers as a poset ordered by divisibility, we realize that the Möbius function can be generalized to sets that are partially ordered and locally finite, as ℕ is. The study of the properties of such a function is of great interest in combinatorics and it relies on the characterization of order structures. I will present this generalization for the subgroup lattice of a finitely generated profinite group and a related conjecture for those which are positively finitely generated (i.e., the probability that some group elements generate the whole group is > 0). This conjecture can be reduced to a problem that only regards finite almost-simple groups. Part of my research involves the study of such a reduction in the case of finite classical groups. In particular, I will introduce some methods and ideas to be applied to PSL(n,q) and PGL(n,q).
Tuesday 24.11.20
Lecture Hall 5L
Martina Conte
Verbal subgroups in profinite groups
Tuesday 01.12.20
Lecture Hall 5L
Karthika Rajeev
Pronilpotent groups I
Tuesday 08.12.20
online
Luca Di Gravina
Pronilpotent groups II
Tuesday 15.12.20
online
Luis de Mendonça
Words of infinite width in pro-p  groups

## Subgroup Growth (programme)

Let G be a finitely generated residually finite group and denote an(G) the number of subgroups of G of index n. The subgroup growth is the study of the asymptotic behaviour of the sequence (an(G))n ∈ ℕ and it turned out that this holds a wealth of information about G. Analogously one can define the sequence for normal subgroups that is denoted by an ( G ). The aim of the seminar is to present 6 self-contained talks around this topic. We will see some results that were not covered during the GRK lectures and some explicit computations.

Tuesday 12.01.21
online
Margherita Piccolo
Normal subgroup growth of free groups
Tuesday 19.01.21
online
Iker de las Heras
Gap Theorem for Pro-p Groups
Tuesday 26.01.21
online
The PSG Theorem
Tuesday 02.02.21
online
Karthika Rajeev
Subgroup growth of Baumslag-Solitar Groups
Tuesday 09.02.21
online
Moritz Petschick
Zeta functions of plane crystallographic groups
Tuesday 16.02.21 (TBC)
online
Djurre Tijsma
Normal subgroup growth of the groups SL1d (Fp[[t]]) for d ∈ {2, 3}