The study of the unit group of the integral group ring ℤG of a finite group G began about 75 years ago with G. Higmans PhD-thesis. Though extensively studied the structure of the unit group is still far from being well understood. It is for example unknown, if the orders of units of finite order in ℤG and the orders of elements of G must coincide. Concerning the units of finite order in ℤG the main open question is the so-called Zassenhaus Conjecture: For any unit u of finite order in ℤG there exist a unit x in the rational group algebra ℚG and an element g ∈ G such that x^-1ux = ±g. We will give an overview on the known results regarding the Zassenhaus Conjecture and related questions, some methods used to attack these questions, mostly involving representation theory of groups, and on some directions of research which seem promising for the near future.

We construct examples of manifolds whose universal covers are large in topological sense but small in homological sense. Such manifolds provide counterexamples to a recent conjecture of A.Dranishnikov. The construction uses Coxeter groups and associated Davis complexes. A motivation to study such manifolds comes from the conjecture of Gromov. It says, that the fact that a smooth closed manifold admits a metric of positive scalar curvature is visible in the topology of its universal cover as a deficiency of the macroscopic dimension. Such a phenomenon takes place when we speak of positive sectional curvature (then the universal cover is compact). In the case of positive scalar curvature no such a result is yet known.

A notion of random surfaces allows one to investigate the properties of a typical surface. In some cases random surfaces can also be used to do existence proofs. That is, sometimes it is easier to prove that the probability that a surface has certain properties is non zero than to directly construct such surfaces. There are multiple models for random surfaces. In this talk I will speak about surfaces obtained from randomly gluing together a given number of triangles. In particular, I will discuss results on short curves on such surfaces and how these relate to the theory of random regular graphs.

Iterated monodromy group (IMG) is a self-similar group associated to every branched covering map f of the 2-sphere (in particularly to every rational map). It was observed that even very simple maps generate groups with complicated structure and exotic properties which are hard to find among groups defined by more "classical" methods. For instance, IMG(z^2+i) is a group of intermediate growth and IMG(z^2-1) is an amenable group of exponential growth. Unfortunately, we still face a lack of general theory which would unify and explain these nice examples.

In 1999 K.S. Brown asked two questions about the connectivity of the coset poset. The coset poset is the collection of all cosets of all proper subgroups of a given group, ordered by inclusion. Brown's first question, "Are there any finite groups G for which the coset poset is contractible?" was answered in the negative by J. Shareshian and R. Woodroofe in 2014. His second question, "For which finite groups G is the coset poset simply connected?" was investigated by D.A. Ramras in 2005. Ramras also showed that the coset poset is contractible if G is not finitely generated.

In my talk I will present new results for coset posets of some finitely presented groups. I will also give a useful tool of working with finite index subgroups of finitely presented groups, starting from a given presentation.

Monod introduced examples of groups of piecewise projective homeomorphisms, which he demonstrated are non amenable despite the fact that they do not contain non abelian free subgroups. In joint work with Justin Moore, I isolated examples of finitely presented groups of piecewise projective homeomorphisms with the same property. I further proved that these groups are also of type F_∞. In recent work with Burillo and Reeves, we investigated the normal subgroup structure of these groups. In this talk I will present a survey of these groups and discuss their striking properties.

An orbit polytope is the convex hull of the orbit of a point under a finite group G ≤ GL(d,ℝ). We consider the possible affine symmetry groups of orbit polytopes. We define a set of "generic" points and show that the non-generic points are the zero-set of some non-zero polynomials. Moreover, the affine symmetry group of a generic orbit polytope is contained up to conjugacy in the affine symmetry group of every other full-dimensional orbit polytope under the same group.

For some few groups G, the affine symmetry group of every orbit polytope is strictly larger than G. This can be determined from the character of G.

We also show that every abstract group that is isomorphic to the full euclidian symmetry group of an orbit polytope, is also isomorphic to the full affine symmetry group of an orbit polytope, with exactly three exceptions: the elementary abelian groups of orders 4, 8 and 16. This answers a question of Babai (1977).

The class of axial algebras was recently introduced by Hall, Rehren, and the speaker, generalizing Majorana algebras of Ivanov. The main motivating example is the real 196884-dimensional Griess-Norton algebra for the sporadic Monster group M. Just like the Monster algebra, general axial algebras often correspond to groups generated by a class of elements of small order, typically, involutions. In the lecture we will discuss recent results about two classes of axial algebras related to groups of 3-transpositions.

An autonomous area preserving diffeomorphism is a time one map of the flow of a (divergence free) verctor field. Every area preserving diffeomorphism is a product of autonomous ones. I will explain that there are diffeomorphisms which are products of arbitrarily many autonomoous diffeomorphisms. The prove uses braids and quasimorphisms. The situation is completely different for all diffeomorphisms. I will briefly discuss the wider picture.

Lattices in the isometry groups of most symmetric spaces are classified, because they are known to be arithmetic (i.e. up to commensurability, they can be obtained as the set of integer matrices in a suitable real form of the relevant group), by celebrated work of Magulis, Corlette, Gromov and Schoen.

In the cases of the isometry groups of real and complex hyperbolic spaces, non-arithmetic lattices are known to exist, but no classification is known. In the complex hyperbolic case, it is not even clear whether there exist infinitely many commensurability classes of non-arithmetic lattices (apart from complex dimension one, where there are uncountably many commensurability classes).

I will present joint work with J. Parker and J. Paupert, that produces a new (finite) class of non-arithmetic lattices in the isometry group of the complex hyperbolic plane.

Ich werde über eine gemeinsame Arbeit mit Heike Sach vortragen – erste Schritte zu etwas, was ein lohnendes Projekt werden könnte.

Es geht um die Bahn Ω = Γp eines Punktes p unter der Operation einer diskreten Isometriegruppe Γ. Mit Hilfe eines geometrisch-endlichen Fundamentalbereichs definieren wir, was wir unter einem „Γ-polyedrischen Bruchstück“ S von Ω verstehen wollen. Eine Permutation f: Ω→Ω heißt stückweise isometrisch, wenn Ω eine Vereinigung solcher „Bruchstücke“ ist, auf denen die Restriktion von f eine isometrisch Einbettung ist. Wir interessieren uns für die Gruppen G(S) aller stückweise isometrischen Permutationen von Γ-polyedrischen Teilmengen S⊆Ω. Die zwei elementarsten Fälle – das Eklidische Gitter ℤⁿ und das PSL(2,Z)-Gitter in der hyperbolischen Ebene – sind ermutigend: 1) Wir können die Endlichkeitslängen von G(S) für viele Euklidische polyedrische Teilmengen S⊆ℤⁿ bestimmen. 2) G(S) ist für jede SL(2,ℤ)-polyedrische Mengen S eine enge Verwandte von Thompson's Gruppe V.

A complex rational function of degree n describes a branched covering of Riemann spheres, whose monodromy group is a transitive permutation group on n letters. The ramification data of this covering yields a specific generating system of this group.

Over arbitrary fields one obtains a similar setting by Galois theory. However, if the base field has positive characteristic, less precise information about these groups is available.

We give some historic and recent examples where combinatorial and group theoretic techniques can be used to answer geometric and number theoretic questions about polynomials and rational functions.

Rational functions with only three critical values, so called Belyi functions, have a nice graph theoretic description. Grothendieck introduced these graphs as dessins d'enfants, however they appeared as Linienzüge in the work of Felix Klein long before.

A contemporary activity is the explicit computation of rational functions from these graphs. We sketch several possible approaches.

A huge group theoretic project is the classification of all the possibilities of monodromy groups of rational functions. We comment on this project, which relies heavily on the classification of the finite simple groups.

Finally, we give an example how an elementary group theoretic lemma settled a recent question about invariant curves in complex analysis.

The McKay conjecture is probably the most accessible counting conjecture in the representation theory of finite groups. It states for every prime p that the number of irreducible characters of p'-degree is determined by the normaliser of its Sylow 2-subgroup.

Isaacs, Malle and Navarro established a new approach to (a general proof of) this conjecture, by reducing it to the now-called inductive conditions on simple groups. Using detailed analysis of Harish-Chandra induction together with Lusztig's parametrisation of characters this approach leads to the proof of the McKay conjecture for the prime 2, hence for characters of odd degree. This is joint work with Gunter Malle.

In his book on trees Serre gives a recipe of how to determine the quotient of a Bruhat-Tits tree of SL₂ over some completed global function field modulo the action of an arithmetic lattice. In case of the rational function field together with Mühlherr and Struyve we developed a method that provides this quotient uniformly without case-by-case analysis for arbitrary valuations.

Unfortunately, I currently do not see any way of generalizing this method to other global function fields as we heavily make use of the fact that the ring of integers corresponding to the chosen valuation is Euclidean.

A PhD student of mine is currently applying this method to SL₃, but there no uniform solution has been found as yet; some classes of examples for valuations of low degree have been worked out.

The key idea of the method is inspired by the Bux-K.-Witzel metric codistance in twin buildings for studying finiteness properties of G(F_q[t]) and G(F_q[t,1/t]), where G is a semisimple group defined over F_q.