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.

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 \( (W,S)\).

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.

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.

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:

1. the minimal number of vertices that have to be deleted such that the clique complex of the remaining graph has homology in dimension \(i\)
2. 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.

Let X be a simply connected finite CW-complex, a classical theorem of Serre says that the homotopy groups of X are all finitely generated. But what happens when X is not simply connected? In this talk, we propose a general way to answer this question by relating it to some cohomological invariants of the fundamental group. We can get a quite satisfactory answer when the fundamental group is a Poincaré Duality group. This is a joint work with Yang Su.

We use the language of formal power series to construct finite state automata generating groups of the form \(A \wr \mathbb{Z}\), where \(A\) is the additive group of a finite commutative ring and \(\mathbb{Z}\) is the integers. We then provide conditions on the units of the ring and the power series which make automata bireversible. This is a joint work with Benjamin Steinberg

Many questions exist concerning the units of a group ring \(\mathbb Z G\) of a finite group \(G\), and many of them are still unanswered. The general philosophy of several of these questions is how ``rigid'' the group \(G\) lies in \({\mathcal U}({\mathbb Z} G)\). One way of assessing this rigidity is by forming amalgamated- or HNN decompositions of \({\mathcal U}({\mathbb Z} G)\). In recent joint work with Andreas B\"{a}chle, Geoffrey Janssens, Eric Jespers and Ann Kiefer, we studied a geometric property, Serre's property (FA), which forms an obstruction for such decompositions. In order to understand for what groups \(G\) the group \({\mathcal U}({\mathbb Z} G)\) has property (FA), we are able to reduce this study to several special linear groups. It appears that for many of these groups a stronger property, Kazhdan's property (T), is already known to hold. On the other hand, for those groups of so called lower rank, the literature is scarce. We were able to show that, for the low rank groups which are relevant to the study of \({\mathcal U}({\mathbb Z} G)\), none of them have property (FA). In this talk we will discuss some of the techniques used to study the geometric property (FA) for these linear groups and how they can be applied to \({\mathcal U}({\mathbb Z} G)\).

There is a classical result of Culler and Morgan saying that minimal semi-simple actions on trees are determined by their marked length spectrum, i.e. by the translation lengths of the group elements. In this talk we want to generalize this result from simplicial trees to cube complexes. Namely, we show that under some assumptions on the cube complex many actions on irreducible CAT(0) cube complexes are determined by their marked length spectrum (w.r.t. the l^1 metric).
The main tool will be a natural cross ratio on some boundary of the cube complex. We show that, given two actions with the same marked length spectrum on X and Y, we find subsets of the boundaries of X and Y where the cross ratios coincide and then show that this subset together with the cross ratio already determines the isomorphism type of the cube complex. Similar as Culler-Morgans result can be used to compactify outer space of free groups, we show that this allows to compactify the Charney-Stambaugh-Vogtmann outer space for irreducible RAAGs.
Joint work with Elia Fioravanti.

A fundamental invariant in the study of hyperbolic groups is their boundaries. The classical Cannon--Thurston map for a closed fibered hyperbolic 3-manifolds relates two such boundaries: It gives a continuous surjection from the boundary of the surface group (a circle) to the boundary of the 3-manifold group (a 2-sphere). Mitra generalized this to all hyperbolic groups with hyperbolic normal subgroups. I will explain why such a generalization to CAT(0) groups with the visual boundary is impossible. Joint work with Beeker--Cordes--Gupta--Stark.

The Post correspondence problem is a decision problem about homomorphisms of free monoids. It was shown to be insoluble by Post in 1946, and has since been much-studied by computer scientists. In a 2014 paper Myasnikov, Nikolaev and Ushakov generalised the problem to groups. In this talk we link the Post correspondence problem for free groups to the problem of finding fixed points of free group endomorphisms. We give partial results, and explain where the difficulty lies in solving the problem for all free groups.

The Dehn function of a finitely presented group $G$ with finite generating set $X$ is a quantitative measure for the difficulty of detecting whether a word in $X$ represents the trivial element in $G$. Dison raised the question if residually free groups admit a uniform polynomial upper bound on their Dehn functions. It is motivated by the existence of uniform polynomial upper bounds on interesting families of residually free groups, such as the Stallings--Bieri groups. In this talk we will show that the answer to Dison's question is negative, by proving that for every $r\geq 3$ there is a subgroup $G_r$ of a direct product of $r$ free groups with Dehn function bounded below by $n^r$. This is joint work with Romain Tessera.

Let \(G\) be a permutation group on a set \(\Omega\). A base for \(G\) is a subset \(B\) of \(\Omega\) such that the pointwise stabiliser of the elements of \(B\) is trivial. There has been a large amount of recent research on the size of a base of a primitive permutation group, culminating in the recent proof of Pyber's Conjecture. At the same time there has been a large amount of work devoted to finding the primitive groups with a base of size two. For such groups we can define the Saxl graph of \(G\) to be the graph with vertex set \(\Omega\) and two elements are joined by an edge if they are a base. I will discuss some recent work with Tim Burness that investigates some of the properties of this graph.