Bielefeld-Münster Seminar
on Groups, Geometry and Topology

Date: 8 Feburary 2016

Venue: Bielefeld University

All talks will take place in V3-201. Coffee will be served in the same room during the breaks.

If you have questions, please contact the organizer.


10:00 UhrDawid Kielak
Thurston and Alexander norms for free-by-cyclic groups.
11:00 UhrCoffee break
11:30 UhrAntoine Beljean
The Leray-Serre Spectral Sequence for Euclidean Buildings
12:30 UhrLunch break
13:30 UhrKai-Uwe Bux
Braided Houghton Groups
14:30 UhrCoffee break
15:00 UhrNils Leder
Quasimorphisms and Bounded Cohomology
15:45 UhrShort break
16:00 UhrHerbert Abels
The topological generating rank of Lie groups
17:00 UhrEnd the of seminar


Dawid Kielak

Thurston and Alexander norms for free-by-cyclic groups.

In the world of knot-complement (and more general 3-manifolds) groups there are two well-established norms on the first cohomology, the Alexander and Thurston norms. We will (briefly) discuss some properties of these norms, and indicate how they can be defined for other classes of groups.

For 3-manifold groups it has been shown by McMullen that the Alexander norm provides a lower bound for the Thurston norm. We will discuss recent development in extending this result to the realm of free-by-cyclic groups (or more general ascending HNN extensions of free groups), as well as a relation between the Thurston norm and the BNS-invariants.

This is joint work with Florian Funke.

Antoine Beljean

The Leray-Serre Spectral Sequence for Euclidean Buildings

The Leray-Serre spectral sequence is a powerfull tool of algebraic topology that one can apply to fibrations. It enables one to express the (co-)homology of the total space in terms of the co-homologies of the base space and of the fiber. The essential ingredient to construct this spectral sequence in the case of a fibration is its homotopy-lifting property.

We propose in this talk to modify the situation: take a Euclidean building as the base space, its direction bundle (which is the union of the spaces of directions of all points in the building, equiped with a nice topology) as the total space, and the space of directions over a point as the fiber over this point. Can one build a spectral-sequence which expresses relations between the homologies of these spaces? Since the homotopy-lifting property is no longer available to construct the spectral sequence, we propose to do it using other nice properties of Euclidean buildings.

Kai-Uwe Bux

Braided Houghton Groups

Houghton groups Hn form a sequence of groups n∈ℕ with increasing finiteness properties. H1 is the group S of permutations of ℕ with finite support. It is not even finitely generated.

H2 is the group of permutations of ℤ that look like x ↦ x + t for x far away from 0. This group is generated by the translation of length one and neighbor transposition: conjugation with the translation provides us with all neighbor transpositions. However, H2 is not finitely presented.

Hn is the group of permutations of the set ℕ×{1,…,n} that are of the form (x,i)↦(x+ti) for very large x. Brown showed that Hn is of type Fn-1 but not of type Fn.

In his thesis, Franz Degenhardt considered braided relatives Hnbr of Houghton groups Hn. As Houghton groups are groups of infinite permutations satisfying a regularity condition, braided Houghton groups consist of infinite braids subject to some restriction.

Degenhardt showed that

He conjectured that braided Houghton groups have the same finiteness properties as their unbraided counter parts.

We relate this question to the problem of higher generation (a notion introduced by Abels and Holz) of braid groups by certain families of subgroups. Using this perspective, we are able to prove Degenhardt's conjecture.

Nils Leder

Quasimorphisms and Bounded Cohomology

I will talk about the article "Quasi-morphisms on Free Groups" by Pascal Rolli. A quasi-morphism on a group Γ is a map f : Γ → ℝ which behaves up to a finite error like a group homomorphism. It is called trivial if it is the sum of a homomorphism and a bounded map. Fixing the group Γ, there is the natural question whether any quasi-morphism is trivial, i.e. comes from a group homomorphism with a bounded perturbation.

Interestingly, the existence of non-trivial quasi-morphisms on Γ is strongly related to the second bounded cohomology group Hb2(Γ; ℝ). In my talk, I want to explain this connection and show how quasi-morphisms on a free group F of rank ≥ 2 can be used to prove that Hb2(F ; ℝ) is infinite-dimensional.

Herbert Abels

The topological generating rank of Lie groups

We define the toplogical generating rank dtop(G) of a topological group G as the minimal number of elements of G which generate a dense subgroup of G. I will present joint work with Gena Noskov, partly in progress, about our attempts to compute the topological generating rank for connected Lie groups. The cases that G is semisimple, where dtop(G) is slightly bigger than one, and that G is nilpotent, have been known. We have a precise formula for the case that G is solvable and bounds for the general case, which we hope are sharp. The reduction steps motivated us to consider - a version of - the Frattini subgroup of G.

Last modified: Mon 04 Jan 2021, 12:05