ARBEITSGRUPPE

GEOMETRIE

In this talk we discuss the dynamic asymptotic dimension for actions of groups on locally compact spaces. Dynamic asymptotic dimension is the dynamical counterpart of the notion of topological dimension which is related to other invariants such as equivariant asymptotic dimension and nuclear dimension.
In the first part of the talk we overview the existing results including the recent rigidity result on the actions of integers on compact spaces. We also discuss the relevance to celebrated conjectures such as that of Baum--Connes, Juan-Pineda--Leary and Farrell--Jones.
In the second part of the talk we discuss new results including rigidity results for actions of virtually cyclic and metacyclic groups. We also include results on the actions by locally compact groups, including those related to non-compact type homogeneous spaces.

In this talk I will show that the commutator of root groups corresponding to nested roots is trivial in a certain class of RGD systems.

Recent results of several authors hint at an expansion of the Freudenthal--Tits magic square, with new rows/columns corresponding to the so-called ternions and sextonions: \(3\)- and \(6\)-dimensional algebras over \(\mathbb{K}\), respectively, very much behaving like the composition algebras used to construct the square, except that they are degenerate.
I will present a class of "degenerate composition algebras" which put these two cases in a general framework, and for the entire second row of the magic square (both the non-split and the split version), I will briefly give a geometric description and axiomatic characterisation of the corresponding varieties, together with their relations to both the non-degenerate versions and to affine buildings.

In this talk we shall present some recent results on the generating rank of classical polar spaces and their polar Grassmannians and the dimension of the Grassmann embedding.

By results of Serre and Bate--Martin--Röhrle, the usual notion of \(\mathbf{G}\)-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of \(\mathbf{G}\), where \(\mathbf{G}\) is a reductive algebraic group.
In this talk, I will introduce a variation of this notion, the so-called relative complete reducibility. I will discuss that relative complete reducibility can also be viewed as a special case of the building-theoretic definition. This is a joint work with Alastair Litterick and Gerhard Röhrle.

A classical result states that the real Lie algebra \(\mathfrak{so}(n)\) admits a presentation via the embedded \(\mathfrak{so}(3)\)-subalgebras along an \(\mathrm{A}_{n-1}\) diagram, similar to the Curtis--Tits theorem and Phan's theorems.
String theorists are interested in studying the corresponding (infinite-dimensional) real Lie algebra \(\mathfrak{k}\) for the diagram \(\mathrm{E}_{10}\). In my talk I will discuss a natural generalization of the classical \(1/2\)-spin representation of \(\mathfrak{so}(n)\) to this Lie algebra \(\mathfrak{k}\). I will exhibit a Cartan--Bott-type periodicity for the images of this representation along the \(\mathrm{E}_n\) series, and I will present a general machinery how to extend this \(1/2\)-spin representation to higher spin representations. It will turn out that all these extended higher spin representations are controlled by a concise Weyl-group-based formula.
The results presented in this talk are based on prior work of Damour, Kleinschmidt, Nicolai and have been obtained jointly with Hainke, Horn, Lautenbacher, Levy in various combinations. The long-term goal is to develop a representation theory of \(\mathfrak{k}\) and to understand the structure of \(\mathfrak{k}\), for instance, whether \(\mathfrak{k}\) is residually finite-dimensional.

We show that the parallelisms described by the title are precisely the \(3\)-dimensional regular parallelisms in the sense of Betten and Riesinger. We point out that there is a problem with the existence proof given by those authors.

Let \(W\) be a Coxeter group. We provide a precise description of the conjugacy classes in \(W\), yielding an analogue of Matsumoto's theorem for the conjugacy problem in arbitrary Coxeter groups.

Ihara introduced a zeta function for co-compact discrete torsion-free subgroups of \(\mathrm{PGL}_2(\mathbb{F})\) for a non-archimedean local field \(\mathbb{F}\) and Serre showed how to view the Ihara zeta function as an object which can be attached to an arbitrary finite graph. In recent work with Anton Deitmar and Minghsuan Kang we have been exploring possible ways to generalise the Ihara zeta function to the non-compact case and to higher dimensions. We present some of the possible approaches to doing this and prove a conjecture of Anton Deitmar about the rationality of the higher-dimensional Ihara zeta function so defined, and also a conjecture made in previous work with Minghsuan Kang about the connection of one such zeta function with an alternating product of Poincaré series.

Extremal geometries are point-line geometries introduced by Arjeh Cohen and collaborators, where points are the extremal elements of specific simple Lie algebras and the lines are some two-dimensional subspaces. However, polar spaces cannot be obtained as extremal geometries.
Together with Hans Cuypers, we describe the so-called inner line ideal geometry. Its points are still the extremal elements, but we give a more general definition of lines using inner ideals. This allows us to obtain both the extremal geometries and polar spaces.

Positively folded galleries arise as images of retractions of buildings onto a fixed apartment and play a role in many areas of maths, such as in the study of affine Hecke algebras, Macdonald polynomials, MV-polytopes, and affine Deligne--Lusztig varieties. In this talk, we will see a new recursive description of the set of end alcoves of folded galleries which are positive with respect to alcove-induced orientations. These results further allow us to find the images of retractions from certain points at infinity. This talk is based on joint work with Elizabeth Milićević, Petra Schwer and Anne Thomas.

We prove an analogue for homogeneous trees and affine buildings of a result of Bourgain on geometric Ramsay theory in Euclidean spaces. In particular, we show that certain configurations of vertices are guaranteed to exist in any set of positive upper density in a homogeneous tree or affine building. This is joint work with M. Björklund and A. Fish.

A metasymplectic space is a geometry with four types of elements, usually called points, lines, planes and hyperlines. In this talk I will give a short overview on how to construct such a space using a Chevalley group of type \(\mathrm{F}_4\). The elements of the constructed geometry will be 'certain' subspaces of a \(26\)-dimensional vector space. Mixing the group will then allow us to obtain a mixed metasymplectic space. Here, however, the hyperlines fail to be subspaces of said vector space.

Given a finitely generated group \(G\), the \(\Sigma\)-invariants of \(G\) consist of geometrically defined subsets \(\Sigma^k(G)\) of the set \(\mathbb{S}(G)\) of all characters \(\chi: G -> \mathbb{R}\) of \(G\). These invariants were introduced independently by Bieri--Strebel and Neumann for \(k=1\) and generalized by Bieri--Renz to the general case in the late 80's in order to determine the finiteness properties of all subgroups \(H\) of \(G\) that contain the commutator subgroup \([G,G]\).
In this talk we determine the \(\Sigma\)-invariants of certain \(S\)-arithmetic subgroups of Borel groups in Chevalley groups. In particular we will determine the finiteness properties of every subgroup \(H\) of the group of upper triangular matrices \(\mathbf{B}_n(\mathbb{Z}[1/p]) < \mathrm{SL}_n(\mathbb{Z}[1/p])\) that contains the group \(\mathbf{U}_n(\mathbb{Z}[1/p])\) of unipotent matrices, where \(p\) is any sufficiently large prime number.

In this talk I will discuss several features of the occurrence of a building of type \(\mathrm{F}_4\) inside a building of type \(\mathrm{E}_7\).

In this talk I follow a geometric approach to the Freudenthal--Tits Magic Square. The geometries in this square are all buildings and parapolar spaces, and it is well known that they can be embedded in a projective space. This construction is however rather algebraic and not always easily accessible if you want to study these geometries with geometric methods. I give a very explicit construction of the geometries in the second and third row as the intersection of quadratic equations in a projective space. I also give a few examples of how I have already used this construction to study these geometries.

We construct a group algebra for every topological group that contains a topology. This group algebra is associated functorially and we call it a weakly complete group algebra that is based on the category of weakly complete vector spaces. In the case that the group is compact, its weakly complete real group algebra contains the group as a pro-Lie group and it contains also a copy of its Lie algebra.

We will describe efforts to classify Tits quadrangles. The focus of our work has been the Tits quadrangles that arise from exceptional groups. We will discuss applications of our results to the study of exceptional groups of relative rank \(1\). This is joint work with Bernhard Mühlherr.

We describe the classification of all \(\tilde{C}_2\)-twin buildings having one residue of exceptional type.