Bielefeld-Münster Seminar 02/15

Bielefeld-Münster Seminar

on Groups, Geometry and Topology

Date: 9 Feburary 2015

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.

The next Bielefeld-Münster Seminar will be on June 12, 2015 in Münster.

Schedule

10:00 Uhr	Catherine Pfaff
	When Outer Space behaves like Teichmuller space (or hyperbolic spaces) & how we can use this to understand Out(F_n)
11:00 Uhr	Coffee break
11:30 Uhr	Rupert McCallum
	Rigidity of Polish group topologies for semi-direct products of Lie groups
12:30 Uhr	Lunch break
14:30 Uhr	Patrick Wegener
	Transitive Hurwitz operation on generators of finite Coxeter groups
15:30 Uhr	Coffee break
16:00 Uhr	Barbara Baumeister
	Mounfang twin buildings and Kac-Moody groups

Abstracts

Catherine Pfaff

When Outer Space behaves like Teichmuller space (or hyperbolic spaces) & how we can use this to understand Out(F_n)

Out(F_n) is one of the most intriguing groups to study because of its natural action on a space, Culler-Vogtmann Outer Space, which both strongly resembles and intricately differs from some of the most well-known and studied spaces, such as Teichmuller space and hyperbolic spaces. In this talk I will present several dynamical results about when Outer Space behaves like these other spaces and explain how we have used them to help understand Out(F_n). This is joint work with Yael Algom-Kfir, Ilya Kapovich, and Lee Mosher.

Rupert McCallum

Rigidity of Polish group topologies for semi-direct products of Lie groups

We'am M. Al-Tameemi, and Robert R. Kallman have recently formulated a criterion for when a semi-direct product of two Polish groups admits just one Polish group topology. They have applied it to proving that semi-direct products of special linear groups and general linear groups with the identity representation are topologically rigid. We present this work and also show how to apply Al-Tammemi and Kallman's criterion to showing that the semi-direct product of SL₂(R) with any of its even-dimensional irreducible real representations is topologically rigid. On the other hand the semi-direct product of SL₂(R) with the adjoint representation is not topologically rigid due to a result of Jacques Tits. The case of odd-dimensional representations of dimension greater than three remains open.

Patrick Wegener

Transitive Hurwitz operation on generators of finite Coxeter groups

Let (W,S) be a finite Coxeter system and T its set of reflections. For reduced T-words we consider the so called Hurwitz operation

(t₁, … , t_n) ↦ (t₁, … , t_i-1, t_i+1, t_i+1 t_i t_i+1, t_i+2, … , t_n).

Conjecturally this operation is transitive on the set of reduced T-words for w ∈ W, if there exists some reduced T-word for w such

W = 〈 t₁ , … , t_n 〉.

In my talk I will sketch the proof for Coxeter groups of type D_n.

Barbara Baumeister

Moufang twin buildings and Kac-Moody groups

Every reductive algebraic group acts nicely on a geometric object, the spherical building, and the simple algebraic and classical groups are characterized by that property as Tits showed.

Ronan and Tits generalized the concept of spherical buildings to that of twin buildings in order to obtain geometric objects for the Kac-Moody groups. Twin buildings are defined in such a way that they possess similar properties to those of spherical buildings; for instance the Moufang property - named after Ruth Moufang - can also be defined for twin buildings.

We will introduce the relevant terminology and discuss whether a Kac-Moody group is characterized by its action on a Moufang twin building. We will then focus on the case of trivalent Moufang twin trees. In that case we are able to completely determine the structure of the "unipotent" subgroups which is also of independent interest.