## About

In this 3-day-long event we will bring together experts in GGT and related areas from Great Britain and Germany, as well as members of the SPP interested in the subject.

#### Monday, December 17:

##### Arc matching complexes and finiteness properties of braided groups (Kai-Uwe Bux)

##### Acylindrically hyperbolic groups with strong fixed point properties (Ashot Minasyan)

##### Outer Space of a free product (Dionysios Syrigos)

##### Simple groups separated by finiteness properties (Stefan Witzel)

##### Property (T) for $\mathrm{Aut}(F_n)$ (Dawid Kielak)

#### Tuesday, December 18:

##### The Lipschitz Metric on Culler-Vogtmann Space (Armando Martino)

##### RAAGs and Stable Commutator Length (Nicolaus Heuer)

##### Partial equality and word problems (Angela Carnevale)

##### Growth of submodules invariant under an endomorphism (Tobias Roßmann)

##### Submodule zeta functions for nilpotent algebras of endomorphisms and their functional equations (Christopher Voll)

##### Symposium Dinner (All participants are invited!)

#### Wednesday, December 19:

##### CAT(0) groups need not be biautomatic (Ian Leary)

##### Cohomological finiteness conditions and soluble groups - a survey (Peter Kropholler)

### Arc matching complexes and finiteness properties of braided groups

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.

### Acylindrically hyperbolic groups with strong fixed point properties

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.)

### Outer Space of a free product

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.

### Simple groups separated by finiteness properties

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.

### Property (T) for $\mathrm{Aut}(F_n)$

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

### The Lipschitz Metric on Culler-Vogtmann Space

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.

### RAAGs and Stable Commutator Length

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 $1/2$.

### Partial equality and word problems

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.

### Growth of submodules invariant under an endomorphism

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.

### Submodule zeta functions for nilpotent algebras of endomorphisms and their functional equations

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.

### CAT(0) groups need not be biautomatic

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.

### Cohomological finiteness conditions and soluble groups - a survey

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.

- Naomi Andrew | University of Southampton
- PD Dr Barbara Baumeister | Bielefeld
- Mr Thomas Brown
- Benjamin Brück | Universität Bielefeld
- Kai-Uwe Bux | Bielefeld
- Angela Carnevale | NUI Galway
- Mr Ho Yiu Chung | University of Southampton
- Jonas Flechsig | University of Münster
- Jelena Grbic | Southampton
- Nicolaus Heuer | University of Oxford
- Christoph Hilmes | University Bielefeld
- Mr Sam Hughes | University of Southampton
- Dr. habil. Dawid Kielak | Bielefeld
- Prof Peter Kropholler | Southampton
- Prof Ian Leary | University of Southampton
- Paula Lins | Universität Bielefeld
- Dr. Keivan Mallahi-Karai | Jacobs University
- Dr Armando Martino | University of Southampton
- Dr Ashot Minasyan | University of Southampton
- Philip Möller | WWU Münster
- Tobias Rossmann | NUI Galway
- Yuri Santos Rego | Universität Bielefeld
- Eduard Schesler | Bielefeld University
- Mima Stanojkovski
- Dr Dionysios Syrigos | University of Southampton
- Elena Tielker
- Motiejus Valiunas | University of Southampton
- Mr Vladimir Vankov | University of Southampton
- Dr. Olga Varghese | WWU Münster
- Christopher Voll | Bielefeld
- PD Dr. Stefan Witzel | Bielefeld University

We will provide a limited number of single rooms for the participants -- if you are interested, please tick the relevant box in the registration form.