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.
Schedule
10:00 Uhr | Dawid Kielak |
Thurston and Alexander norms for free-by-cyclic groups. | |
11:00 Uhr | Coffee break |
11:30 Uhr | Antoine Beljean |
The Leray-Serre Spectral Sequence for Euclidean Buildings | |
12:30 Uhr | Lunch break |
13:30 Uhr | Kai-Uwe Bux |
Braided Houghton Groups | |
14:30 Uhr | Coffee break |
15:00 Uhr | Nils Leder |
Quasimorphisms and Bounded Cohomology | |
15:45 Uhr | Short break |
16:00 Uhr | Herbert Abels |
The topological generating rank of Lie groups | |
17:00 Uhr | End the of seminar |
Abstracts
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 BuildingsThe 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 GroupsHoughton groups H_{n} form a sequence of groups n∈ℕ with increasing finiteness properties. H_{1} is the group S_{∞} of permutations of ℕ with finite support. It is not even finitely generated.
H_{2} 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, H_{2} is
H_{n} is the group of permutations of the set ℕ×{1,…,n} that are of the form (x,i)↦(x+t_{i}) for very large x. Brown showed that H_{n} is of type F_{n-1} but not of type F_{n}.
In his thesis, Franz Degenhardt considered braided relatives H_{n}^{br} of Houghton groups H_{n}. 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
- H_{1}^{br} is not finitely generated.
- H_{2}^{br} is finitely generated but not finitely presented.
- H_{3}^{br} is finitely presented but not of type F_{3}.
- H_{n}^{br} is of type F_{3} for n ≥ 4.
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 CohomologyI 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 H_{b}^{2}(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 H_{b}^{2}(F ; ℝ) is infinite-dimensional.
Herbert Abels
The topological generating rank of Lie groupsWe define the toplogical generating rank d_{top}(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 d_{top}(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.