**Bielefeld-Münster Seminar**

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 SL_{2}(R) with any of its even-dimensional irreducible real representations is topologically rigid. On the other hand the semi-direct product of SL_{2}(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

(t_{1}, … , t_{n}) ↦ (t_{1}, … , 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 = 〈 t_{1} , … , 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.