Brin and Dehornoy introduced braided versions of Thompson groups. To prove that these groups are of type $F_\infty$, arc matching complexes are the right tool. A matching in a given graph $\Gamma$ is a collection of pairwise disjoint edges. The matchings in $\Gamma$ form a simplicial complex. An arc matching is a way to draw a matching onto a fixed surface that contains the vertices of $\Gamma$ as marked points. We shall investigate vanishing of higher homotopy groups of arc matching compleses. New examples of arc matching complexes yield further applications to finiteness properties of braided groups. In particular, we can verify a conjecture of Degenhardt about the series of braided Houghton groups. As another application, we obtain highly generating families (in the sense of Abels-Holz) of subgroups in braid groups.

The concept of an acylindrically hyperbolic group, introduced by D. Osin, generalizes hyperbolic and relatively hyperbolic groups, and includes many other groups of interest: $\mathrm{Out}(F_n)$, $n>1$, most mapping class groups, directly indecomposable non-cyclic right angled Artin groups, most graph products, groups of deficiency at least $2$, etc. Roughly speaking, a group $G$ is acylindrically hyperbolic if there is a (possibly infinite) generating set $X$ of $G$ such that the Cayley graph $\Gamma(G,X)$ is hyperbolic and the action of $G$ on it is "sufficiently nice". Many global properties of hyperbolic/relatively hyperbolic groups have been also proved for acylindrically hyperbolic groups. In the talk I will discuss a method which allows to construct a common acylindrically hyperbolic quotient for any countable family of countable acylindrically hyperbolic groups. This allows us to produce acylindrically hyperbolic groups with many unexpected properties. (The talk will be based on joint work with Denis Osin.)

Culler- Vogtmann space has been used successfully for the study of automorphisms of free groups. Later, Guirardel and Levitt generalised this space for general free products. In this talk, I will describe how classical tools in the study of $\mathrm{Out}(F_n)$, like the Lipschitz metric and the train track representatives, can be generalised in the more general context, but I will also explain the differences of the classical and the general case. Finally, I will talk about some recent results of this area and some on-going projects.

I will speak about joint work with Rachel Skipper and Matt Zaremsky. We prove that for every n there exists a simple group that is of type $F_{n-1}$ but not of type $F_n$. Since finiteness properties are quasi-isometry invariants, we in particular obtain the second known infinite family of quasi-isometry classes of simple groups. The first such family consists of Kac--Moody groups and is due to Pierre-Emmanuel Caprace and Bertrand Rémy. Our examples are Nekrashevych groups, that is, they are built out of a Higman--Thompson group and a self-similar group. We show that under certain circumstances the Nekrashevych group inherits (virtual) simplicity from the Higman--Thompson group and the finiteness properties from the self-similar group.

I will discuss a recent proof of property (T) for $\mathrm{Aut}(F_n)$, due to M. Kaluba, P. Nowak, and myself.

The study of $\mathrm{Out}(F_n)$ involves understanding both its action on Culler-Vogtmann Space, as well as certain normal forms called (relative) train track maps. Due to a result of Bestvina, the Lipschitz metric provides a link between these two approaches. We outline some of the recent results in the area; the isometry group with respect to the metric, a new solution of the conjugacy problem for irreducible automorphisms and some future directions for research.

Stable commutator length (scl) is a well established invariant of group elements $g$ (write $\mathrm{scl}(g)$) and has both geometric and algebraic meaning. Many classes of "non-positively curved" groups have a gap in stable commutator length: This is, for every non-trivial element $g$, $\mathrm{scl}(g) > C$ for some $C > 0$. This gap may be thought of as an algebraic injectivity radius and may be found in hyperbolic groups, Baumslag-solitair groups, free products and Mapping Class Groups. However, the exact size of this gap usually unknown, which is due to a lack of a good source of quasimorphisms. In this talk I will construct a new source of quasimorphisms which yield optimal gaps and show that for Right-Angled Artin Groups and their subgroups the gap of stable commutator length is exactly $1/2$.

I will present some recent results on (partial) equality and word problems for finitely generated groups. In particular, I will focus on (un)solvability of generic versions of these two problems with respect to classical and Banach densities. This is joint work with Matteo Cavaleri.

Finitely generated torsion-free nilpotent groups of maximal class have quadratic normal subgroup growth. I will explain how this result follows from analytic properties of zeta functions enumerating those subgroups of a free abelian group which are invariant under a given endomorphism. I will also discuss applications to the study of subobject growth over power series rings.

I will review recent work on Dirichlet-type generating series enumerating lattices which are invariant under nilpotent algebras of endomorphisms. Examples of such series include the ideal zeta functions enumerating ideals in nilpotent (Lie) rings. Under certain homogeneity conditions on the algebra of endomorphisms, these series satisfy local functional equations. I will explain how these kind of counting problems may be interpreted geometrically in terms of affine Bruhat-Tits buildings and p-adic integration.

Ashot Minasyan and I have found a family of groups that are CAT(0) but not biautomatic, resolving a well-known question. These groups also give a negative answer to a question of Wise. I will talk about the groups, why they have the claimed properties, and some questions that remain open.

A look at the classical finiteness and cohomological finiteness conditions and what we know about them for the class of soluble groups. A finiteness condition in group theory is any property of groups that holds for all finite groups. A homotopical finiteness condition is any property of groups that holds for all groups that admit a finite Eilenberg-Mac Lane space. Torsion-freeness is an example of the latter and residual finitenss is an example of the former. The $FP_\infty$ property is an example that is a finiteness condition of both kinds.