3 April 2019

Matteo Vannacci

**Pro-p groups with quadratic cohomology and generalised p-RAAGs**

**14:15 - 15:45 **in** V4-119:** A pro-p group is said to be *quadratic* if its F_p-cohomology algebra is generated in degree one and its relations come from degree two elements. The main example of a quadratic group is the absolute Galois group of a field, so the study of quadratic groups could lead to new insight into the mysterious class of absolute Galois groups. In this talk I will define a new class of pro-p groups called *generalised p-RAAGs* as a natural generalisation of Right Angled Artin Groups. These groups give uncountably many new examples of quadratic pro-p groups.

10 April 2019

Patrick Wegener

**Diagrammatics of Reflection and Artin Groups**

**14:15 - 15:45 **in** V4-119:** For a finite Coxeter system \((W,S)\) of rank \( n\) with set of
reflections \(T\), we associate to every subset \( \{ t_1, \ldots, t_n \}
\subseteq T\) with \( \langle t_1, \ldots , t_n \rangle = W\) a
"Dynkin-like" diagram and show that it encodes a presentation of \( W\)
(like the Dynkin diagram does). This extends previous work of
Cameron-Seidel-Tsaranov as well as Barot-Marsh. We further outline an
explicit construction of these diagrams and connect them with cluster
algebras. At the end of my talk I will discuss to what extent these
diagrams also encode a presentation of the Artin group associated to
\( (W,S)\).

17 April 2019

Leo Margolis

**Orders of units in integral group rings**

**14:15 - 15:45 **in** V4-119:** Since the study of the unit group of integral group rings \(\mathbb{Z}G\) of a finite group \(G\) began with G. Higman's thesis in 1940 many conjectures have been put forward regarding the finite subgroups of units in \(\mathbb{Z}G\). The strongest of those, such as the Isomorphism Problem or the Zassenhaus Conjectures, gave rice to fascinating mathematics, but turned out to be wrong in general, with counterexamples in the class of solvable groups. On the other hand the strongest possible expectations one can have concerning possible orders of units in \(\mathbb{Z}G\) are known to hold for solvable groups.
Namely, call a unit normalized if its coefficients sum up to one. Then the best possible statement one can hope for regarding orders of units in \(\mathbb{Z}G\) is that there is a normalized unit of order \(n\) in \(\mathbb{Z}G\) if and only if there is a group element of order \(n\) in \(G\). The question if this holds for any \(G\) is known as the Spectrum Problem. The weaker form of the question which one obtains by replacing \(n\) by the product of two distinct primes is known as the Prime Graph Question. The Spectrum Problem is known to have a positive answer for solvable groups and for the Prime Graph Question also a reduction theorem, to almost simple groups, has been obtained.
I will present a result which states that if \(p\) and \(q\) are primes and the Sylow subgroup of \(G\) is cyclic of order \(p\) then \(\mathbb{Z}G\) contains a normalized unit of order \(pq\) if and only if \(G\) contains an element of order \(pq\). The main ingredient of the proof is the description of modules for blocks of defect 1 and their visualisation using Brauer trees. This directly settles the Prime Graph Question for most sporadic groups.
This is joint work with M. Caicedo.

24 April 2019

Sarah Rees

**Rewriting in Artin groups**

**14:15 - 15:45 **in** V4-119:** The class of Artin groups is easy to define, via presentations,
which have the form

\(
\langle x_1,x_2,\cdots,x_n \mid \overbrace{x_ix_jx_i\cdots}^{ m_{ij}}= \overbrace{x_jx_ix_j \cdots}^{m_{ij}}, i\neq j \in \{1,2,\ldots,n\}\rangle
,\, m_{ij} \in {\mathbb{N}} \cup \{ \infty\}, m_{ij} \geq 2.
\)

But it contains a variety of groups with apparently quite different properties.
For the class as a whole, many problems remain open, including the word problem;
this is in contrast to the situation for Coxeter groups, which arise as quotients of Artin groups.
I'll discuss what is known about rewrite systems for Artin groups, and evidence
for the possibility of a general approach to rewriting in these groups.
I'll give some general background,
starting at work of Artin, then Garside, Deligne, Brieskorn-Saito,
then move on, via Appel-Schupp, to
very recent work, by myself and Derek Holt (and sometimes Laura Ciobanu), by Eddy Godelle and
Patrick Dehornoy, also by Blasco, Huang-Osajda.

8 May 2019

Volkmar Welker

**Higher dimensional connectivity versus minimal degree of random graphs and minimal free resolutions**

**14:15 - 15:45 **in

** V4-119:** We study the clique complex, i.e. simplices are subsets of the vertex sets that form a complete subgraph of a random graph sampled from the
Erdős-Renyi model. Motivated by applications in the study of
minimal free resolutions of the Stanley-Reisner ring of the clique
complex, we study for \( i \geq 0\) two invariants:

- the minimal number of vertices that have to be deleted such that the
clique complex of the remaining graph has homology in dimension \(i\)
- the minimal number of vertices that have to be deleted such that an \(i\)-simplex in the clique complex has empty link.

Random graph theory says that the two invariants coincide for \(i = 0\) and all (Erdős-Renyi) probability regimes. We show the same for a middle density regime in case \(i=1\), one inequality for all \(i\) and conjecture equality in general.

15 May 2019

TBA

**TBA**

**14:15 - 15:45 **in** V4-119:**

22 May 2019

**FREIER PLATZ**

29 May 2019

Doryan Temmerman

**TBA**

**14:15 - 15:45 **in** V4-119:**

5 June 2019

Jonas Beyrer

**TBA**

**14:15 - 15:45 **in** V4-119:**

12 June 2019

**FREIER PLATZ**

19 June 2019

Giles Gardam

**Part I of a double feature on one-relator groups**

**14:00 - 14:55 **in** V4-119:** Double feature on one-relator groups. We start at 2pm sharp!

19 June 2019

Alan Logan

**Part II of a double feature on one-relator groups**

**15:05 - 16:00 **in** :**

26 June 2019

Claudio Llosa Isenrich

**TBA**

**14:15 - 15:45 **in** :** TBA

3 July 2019

**FREIER PLATZ**

10 July 2019

**FREIER PLATZ**