Let G be a reductive algebraic group acting on an affine algebraic variety X. In geometric invariant theory, one studies the properties of orbits and stabilizers for the action, and investigates the geometry of the quotient variety X/G. Luna introduced the notion of an étale slice in 1973; roughly speaking, it is a closed subvariety S of X that is transversal to the orbits of the action, and the geometry of S is closely related to the geometry of X/G. Many consequences flow from the existence of an étale slice.

The theory of étale slices works well when the ground field has characteristic zero. In positive characteristic, however, slices can fail to exist. In this talk I will discuss recent work with Bate and Geranios, in which we show that some of the nice consequences of slice theory hold even when the slices themselves are absent. A key ingredient is Serre's notion of G-complete reducibility for subgroups of G.

This talk concerns the long-standing problem of understanding subgroups of simple algebraic groups over algebraically closed fields. I will discuss two aspects of this problem, which lend themselves to very different approaches: Classifying finite simple subgroups, and classifying connected reductive subgroups. In particular, I will discuss an ongoing project with Adam Thomas (Bristol) to classify so-called `non-G-completely reducible' reductive subgroups of simple algebraic groups of exceptional type.

Relatively hyperbolic groups were introduced by Gromov (1987). It is known (see Hrushka (2010)) that essentially different definitions given by Gromov, Bowditch, and Osin are coincide in the case where the ambient group is countable and the peripheral subgroups are infinite. To stress the importance of this subject, we distinguish the following two facts:

1) Limit groups are hyperbolic relative to a collection of representatives of conjugacy classes of maximal noncyclic abelian subgroups (Dahmani).
2) There are finitely generated groups different from ℤ₂ with exactly two conjugacy classes of elements (proved by Osin with the help of small cancellation theory developed for relatively hyperbolic groups).

In the first part of my talk I will present the combinatorial definition of Osin and give some useful statements which help to work in this area. In the second part I will explain new results concerning the conjugacy of subgroups in relatively hyperbolic groups. This is a joint work with K.-U. Bux.

The conjugacy class sizes form a natural set of integers that can be associated to a finite group and much research has been carried out since the work of Sylow about how this set can be used to determine finite group structure. In this talk I will present two different types of result. First I will discuss finite groups in which, for a prime p, all p-elements have conjugacy class size not divisible by p. In particular, I will highlight when this condition implies the group has an abelian Sylow p-subgroup; this has ties to Brauer's problem of using the character table to determine if a group has an abelian Sylow p-subgroup. Second I shall introduce the notion of vanishing conjugacy classes, which is a property that is defined using the character table. I will then present some new results which show that certain structural properties deduced from conjugacy class sizes can in fact be obtained by only considering the vanishing conjugacy class sizes.

A profinite group G is said to have finite lower rank if there exist an integer r and a base of neighbourhoods of the identity in G made of r-generated open subgroups; the smallest such r is called the "lower rank" of G. This invariant has several interesting properties, but it is in general very difficult to compute and exhibiting new profinite groups of finite lower rank is of considerable interest.

I will start with a gentle introduction of general profinite groups and hereditarily just infinite groups. In the end I will present the following result (joint work with B. Klopsch): the direct product of finitely many pairwise non-commensurable hereditarily just infinite profinite groups of finite lower rank has finite lower rank.

I will explain some background about defining characteristic representations of reductive algebraic groups and the finite groups of Lie type they contain (including a famous conjecture by Lusztig and its current status). Then I will talk about the explicit characters and representations of such groups which I have computed. I will sketch the range of examples accessible to computations and some ideas of the underlying methods.

Groups acting properly and cocompactly on nonpositively curved metric spaces enjoy several nice properties. In the case of braid groups with at most 6 strands, I will explain their action on a CAT(0) simplicial complex (joint work with D.Kielak and P.Schwer). On the other hand, I will show that only the 3-strand braid group can act cocompactly on a CAT(0) cube complex. I will show how to extend this last result to other Artin groups and mapping class groups.

Finite dimensional Nichols algebras have been intensively studied and classified in the last ten years. One of the most important tools in this context is the so-called Weyl groupoid of the Nichols algebra. The Weyl groupoid could also be a good starting point to study infinite dimensional Nichols algebras: In this talk, we will discuss first steps towards a classification of affine Nichols algebras of diagonal type. Some of the required results are of purely combinatorial nature.

The common first order theory T_fg that is shared by all non-abelian free groups is stable in the model theoretic sense. This means that we can apply the very general notion of forking independence to ask whether a set of elements in a free group is independent or not.

The aim of this talk is to use this independence notion in order to look at the analogues of primitive elements in non-standard models of T_fg, i.e. groups that share the same theory as free groups but are not free themselves. After giving a brief introduction on the model theoretic backgrounds, I will present the concept of hyperbolic towers introduced by Z. Sela to get a translation of the problem in the language of geometric group theory. Using these hyperbolic towers, I will give a construction of models of T_fg that contain sets which are analogue to bases in free groups in the sense that they satisfy exactly the same first order formulas. However, I will show that in contrast to free groups, those analogues can have arbitrarily large differences in their sizes.

Tropicalizations of arrangement complements turn out to be rational polyhedral fans whose link at the origin is homeomorphic to the order complex of the respective intersection lattice. On the level of matroids, the so-called Bergman fans are discrete-geometric constructions that allow to recover the matroid. Proving the latter requires an intriguing mix of discrete-geometric and tropical techniques. To explain the result, I will review relevant material from both matroid theory and tropical geometry on the way.

The Higman group was constructed as the first example of a finitely presented infinite group without non-trivial finite quotients. Despite this exotic behaviour, I will describe striking similarities with mapping class groups of hyperbolic surfaces, outer automorphisms of free groups and special linear groups over the integers. The main object of study will be the cocompact action of the group on a CAT(0) square complex naturally associated to its standard presentation. This action, which turns out to be intrinsic, can be used to explicitly compute the automorphism group and outer automorphism group of the Higman group, and to show that the group is both Hopfian and co-Hopfian. A surprisingly stronger result actually holds: Every non-trivial morphism from the Higman group to itself is an automorphism.

I will present a construction of finitely generated groups containing isometrically embedded expanders. Such groups have many exotic properties. For instance, they do not embed coarsely into a Hilbert space, and the Baum-Connes conjecture with coefficients fails for them. The construction allows us to provide the first examples of groups that lack property A (i.e., they are not exact) but are still coarsely embeddable into a Hilbert space. Better still, these groups act properly on CAT(0) cubical complexes. To end with, I will also present some further applications of the main construction concerning aspherical manifolds and the asymptotic dimension.