Bielefeld-Münster Seminar on Groups, Geometry and Topology

Friday, July 28, 2023.

All lectures take place in V2-213 in Bielefeld.

10:00 Elisa Hartmann (Bielefeld) A metric approach to topological Σ-invariants
Abstract: Finiteness properties of groups like being of type Fm and of type FPm can be studied via the Rips filtration using Brown's criterion and a result by Alonso. For Σ-invariants of a group there also exists a version of Browns criterion which tells via the Rips-filtration of a submonoid if a character belongs to the Σ-invariants. Recently Kochloukova extended Σ-theory to groups equipped with a locally compact Hausdorff topology. We develop both a homotopical and a homological version of Σ-theory using the Rips filtration assigned to the word metric according to a compact generating set. This agrees with classical Σ-theory which appear if the group is equipped with the discrete topology and with the Σ-theory by Kochloukova. In this talk we present more general versions of famous results in classical Σ-theory. There are also some results which are new to classical Σ-theory which could not yet been extended to the continuous setting. Like a new criterion for higher homological Σ-invariants which can be realized as an algorithm. There also is an algorithm that computes higher homotopical Σ-invariants.

11:00 Coffee break

11:30 Olga Varghese (Düsseldorf) On quotients of Coxeter Groups
Abstract: For a group G we denote by F(G) the set of isomorphism classes of finite quotients of G. We address the following question: Let W be a Coxeter group. Given a finitely generated residually finite group G with F(G)=F(W), what can be said about G?

12:30 Lunch break

14:00 Lara Beßmann (Münster) Automatic continuity for universal groups
Abstract: Originally universal groups have been introduced by Burger and Mozes as automorphism groups of regular trees. Later these groups have been generalised to right-angled buildings by De Medts, da Silva, and Struyve. Equipped with a group topology, universal groups provide interesting examples with nice properties. We introduce the notion of universal groups and discuss some properties. The question of automatic continuity asks for the continuity of given abstract group homomorphisms between topological groups. We study this question for abstract group homomorphism to universal groups and present properties ensuring continuity.

15:00 Coffee break

15:30 Ilaria Castellano (Bielefeld) The rational discrete cohomological dimension of totally disconnected locally compact groups
Abstract: The general structure theory of locally compact Hausdorff groups is a well-established topic in mathematics. With the solution to Hilbert’s 5th Problem on characterising Lie groups, the connected locally compact groups are considered to be well-understood and, therefore, the missing piece is a structure theory for totally disconnected locally compact (= t.d.l.c.) groups. The archetypal example of a t.d.l.c. group is the automorphism group of a locally finite regular tree. The study of t.d.l.c. groups takes inspiration from both Lie groups and discrete groups, but this talk will focus on their similarities to discrete groups. Given a t.d.l.c. group, the rational discrete cohomology has been introduced with the motivation to obtain an interpretation of the low-dimensional cohomology groups by analogy to discrete groups. Such a cohomology theory produces cohomological invariants that can be used to investigate the structure of t.d.l.c. groups and, in this talk, we will focus on the rational discrete cohomological dimension. In this talk I will report on recent results and discuss work in progress and open problems.