Tea Seminar of our group

Place and time:

Semester:		Winter term 2022
Zeit:		Mo 14:15 - 15:15
Ort:		SRZ 214 (hybrid)

Inhalt:

In this seminar, guests and members of our group present their research, or topics of interest to us. Presently we use a hybrid format for the seminar (in-person and zoom). In case you are interested, please contact Dr. Sathaye

Talks:

24.10.2022 Jeffrey Carlson (Imperial College London) Equivariant formality of isotropy actions and products of spheres

Abstract: For best-studied class of homogeneous spaces G/H with G > H compact, connected Lie groups, those for which rank(G) = rank(H), it is well known that the standard left ("isotropy") action of H on G/H is equivariantly formal, meaning every rational cohomology class on G/H lifts to a class in Borel H-equivariant cohomology.

Moving to the case rank(G) - rank(H) = 1, we give a characterization of pairs (G,H) such that the isotropy action is equivariantly formal, via a sequence of reductions ending with pairs such that G/H has the rational homotopy type of a product of spheres. The irreducible such pairs are for the most part already classified in works of Kramer, Wolfrom, and Bletz-Siebert, which require only mild extension to handle the cases we asre interested in and then reduce the entire problem to a verification of finitely many cases. This work is joint with Chen He.

31.10.2022 Konstantin Andritsch (ETH Zürich) Bounded Cohomology and Bounded Acyclicity of groups acting on Cantor sets

Abstract: Bounded Cohomology is a functional-analytic analogue of ordinary cohomology. There are various applications in different fields from rigidity theory, actions on the circle up to the geometry of manifolds. However, although bounded cohomology can be very useful, the problem lies in calculating it since it is in general very hard to compute.

I will talk about recent developments in calculating bounded cohomology of discrete groups. In particular, I am going to elaborate on the bounded acyclicity, meaning the vanishing of bounded cohomology with real coefficients in non-trivial degree, of the full homeomorphism group of the Cantor set and Thompson's group V.

14.11.2022 José Pedro Quintanilha (University of Bielefeld) An introduction to Sigma-invariants

Abstract: The features of a group being finitely generated or finitely presented are well-known to be, respectively, the n=1 and n=2 cases of the property F_n: a group G is said to be of type F_n if it admits a K(G,1) with finite n-skeleton. Towards the end of the last century, the question of when such finiteness conditions descend to subgroups of G led to the discovery of the homotopical Sigma-invariants Σⁿ(G) and their homological counterparts Σⁿ(G;A) (for A a ZG-module). Intuitively, Σⁿ(G) can be thought of as the set of group homomorphims χ : G → R for which G is "of type F_n in the direction of χ", with the classical property F_n being equivalent to 0 ∈ Σⁿ(G).

In my talk I will give an introduction to the theory of Sigma-invariants, with particular focus on Σ¹, mentioning some of their applications in group theory and topology. If time permits I will also say a few words on the more recent efforts to extend these notions to locally compact groups.

5.12.2022 François Thilmany (UC Louvain) Using hyperbolic Coxeter groups to construct highly regular expander graphs

Abstract: A graph X is defined inductively to be (a₀, . . . , a_n−1)-regular if X is a₀-regular and for every vertex v of X, the sphere of radius 1 around v is an (a₁, . . . , a_n−1)-regular graph. A family F of graphs is said to be an expander family if there is a uniform lower bound on the Cheeger constant of all the graphs in F.

After briefly (re)introducing Coxeter groups and their geometries, we will describe how they can be used to construct very regular polytopes, which in turn can yield highly regular graphs. We will then use the super-approximation machinery, whenever the Coxeter group is hyperbolic, to obtain the expansion of these family of graphs. As a result, we will obtain interesting infinite families of highly regular expander graphs, some of which are related to the exceptional groups. The talk is based on work joint with Conder, Lubotzky and Schillewaert.

12.12.2022 Mireille Soergel (ETH Zürich) A generalized Davis-Moussong complex for Dyer groups

Abstract: One common feature of Coxeter groups and right-angled Artin groups is their solution to the word problem. I will introduce Dyer groups, as a class of groups sharing this feature. Which other properties do Dyer groups share with Coxeter groups and right-angled Artin groups? I will give a first answer to this question. Finally I hope to explain how to construct actions of Dyer groups on CAT(0) spaces that extend those of Coxeter groups on Davis–Moussong complexes and those of right-angled Artin groups on Salvetti complexes.