The Euler-Poincaré characteristic of a discrete group is an important (but also quite mysterious) invariant. It is usually just an integer or a rational number and reflects many quite significant properties. The realm of totally disconnected locally compact groups admits an Euler-Poincaré characteristic: surprisingly it is no longer just a number but it is a rational multiple of a Haar measure. A key source of totally disconnected locally compact groups consists of the automorphism groups of locally finite graphs. Since the computation of the Euler-Poincarè characteristic of the automorphism group is quite gentle when the graph is a (coloured) regular tree, I plan to use such groups as leading examples throughout the talk. In particular, I will show that the Euler-Poincaré characteristic turns out to be miraculously related to a zeta-function and, because of it, I plan to conclude the talk by opening more questions than those I was able to address. Based on a joint work with Gianmarco Chinello and Thomas Weigel.

Using ideas from 3-manifolds, Hatcher--Wahl defined a notion of automorphism groups of free groups with boundary. We study their Torelli subgroups, adapting ideas introduced by Putman for surface mapping class groups. Our main results show that these groups are finitely generated, and also that they satisfy an appropriate version of the Birman exact sequence.

Let $M_n$ denote the connect sum of $n$ copies of $S^2\times S^1$. Laudenbach showed that the mapping class group $Mod(M_n)$ is an extension of the group $Out(F_n)$ by $(\mathbb{Z}/2)^n,$ where the latter group is the 'sphere twist' subgroup of $Mod(M_n)$. In joint work with N. Broaddus and A. Putman, we have shown that in fact this extension splits. In this talk, we will describe the splitting and highlight some simplifications of Laudenbach's original proof that arise from our techniques.

A matroid is a combinatorial object based on an abstraction of linear independence in vector spaces and forests in graphs. I will discuss how matroid theory interacts with algebra via the so-called von Staudt constructions. These are combinatorial gadgets to encode polynomials in matroids. A main application is concerned with generalized matroid representations over division rings, matrix rings and probability space representations together with their relation to group theory.

Systolic complexes were introduced by Januszkiewicz and Swiatkowski as a combinatorial form of non-positive curvature. We want to construct systolic complexes using Cayley graphs. We give conditions on a presentation of a group, which imply that the flag complex of the Cayley graph is well-defined and systolic. The main applications of the result concern Garside groups and Artin groups. We will give a classification of the Garside groups whose presentation using the simple elements as generators satisfy our conditions.

Flag Hilbert–Poincaré series arise in the context of local Igusa zeta functions associated to hyperplane arrangements and are connected to other seemingly different enumeration problems in algebra and geometry. We study a coarsening of these series in the case of real hyperplane arrangements by applying geometric and combinatorial tools related to their chambers. In this case, we prove that the numerators of these series are coefficient-wise bounded below by the Eulerian polynomial and equality holds if and only if all chambers are simplicial. This generalizes a recent theorem of Maglione–Voll. This is joint work with Lukas Kühne.

We associate to any matroid a motivic zeta function. If the matroid is representable by a complex hyperplane arrangement, then this coincides with the motivic Igusa zeta function of the arrangement. Although the motivic zeta function is a valuative invariant which is finer than the characteristic polynomial, it is not obvious how one should extract meaningful combinatorial data from the motivic zeta function. One strategy is to specialize to the topological zeta function. I will survey what is known about these functions and, time-permitting, discuss some open questions.

The conjugacy growth function of a finitely generated group counts the conjugacy classes that one can spell with n generators (just as the standard growth function counts group elements). Like standard growth, the asymptotic behaviour of conjugacy growth is a group invariant. The associated generating function (the conjugacy growth series) gives a finer measure of the growth of conjugacy classes. I will give an overview of what is known about conjugacy growth in both senses, and present some recent work on a certain family of (virtually) class 2 nilpotent groups, the so-called higher Heisenberg groups.

It is well known that a group of type E7 contains many conjugacy classes of groups of relative type F4. In this talk we will look at the geometrical counterpart and consider certain inclusions of related geometrical structures (buildings and their Grassmannians, as will be explained). (Joint work with N.S.N. Sastry and H. Van Maldeghem.)

GGS-groups are groups that admit faithful actions on regular rooted trees, and have strong self-similarity features. We will investigate the asymptotic distribution of irreducible complex representations of GGS-groups. I will prove that a large class of GGS-groups have polynomial representation growth and provide a bound for the degree of growth.

Note the change of date! The room for the streaming is T2-213.

Sigma invariants are certain subsets of the character sphere associated to a group of type $F_\infty$ that can be used to determine the homotopical and homological finiteness properties of coabelian subgroups. We will describe a sufficient condition for a character to be in the $n$-th Sigma invariant for even Artin groups of FC-type. We will also explain how in some particular cases this condition is necessary. This is a joint work with Rubén Blasco and José Ignacio Cogolludo.

Let $G$ be a finite insoluble group with soluble radical $R(G)$. The vertices of the soluble graph of $G$, denoted $\Gamma_S(G)$, are labelled by the elements in $G\setminus R(G)$, with $x$ and $y$ adjacent if they generate a soluble subgroup of $G$. This is a natural generalisation of the widely studied commuting graph of $G$. In this talk I will report on recent joint work with Andrea Lucchini and Daniele Nemmi, which establishes several new results on $\Gamma_S(G)$. Our main result states that this graph is always connected, with diameter at most $5$. We can construct examples with diameter $4$, so our upper bound on the diameter is close to best possible. I will explain some of the main ideas and I will present a number of open problems.