**Bielefeld-Münster Seminar**

Date: January 28, 2019

Venue: Bielefeld University

The talks will take place in V2-210. Coffee will be served in the same room during the breaks.

### Schedule

10:00 Uhr | Benjamin Brück |

Free factors at infinity | |

11:00 Uhr | Coffee break |

11:30 Uhr | Yuri Santos |

Soluble matrix groups and their finiteness lengths | |

12:30 Uhr | Lunch break |

14:00 Uhr | Olga Varghese |

Abstract group homomorphisms between locally compact groups and discrete groups | |

15:00 Uhr | Coffee break |

15:30 Uhr | Nils Leder |

Property FA for Automorphism Groups of small Graph Products | |

16:30 Uhr | End the of seminar |

### Abstracts

#### Benjamin Brück

*Free factors at infinity*

Borel and Serre constructed a compactification of symmetric spaces known
as the Borel-Serre bordification. They showed that the topology at
infinity of this structure can be described combinatorially by a
rational Tits building. Outer space is a contractible simplicial complex
on which Out(F_{n}), the outer automorphism group of the free group, acts
cocompactly and with finite stabilisers. It is often seen as an
Out(F_{n})-analogue of symmetric spaces.

There is also an Out(F_{n})-complex which shares many properties with
buildings of type A_{n}, namely the free factor complex defined by Hatcher
and Vogtmann. In this talk, I will explain how the free factor complex
can be seen as "sitting at infinity" of Outer space, obtaining an
analogue of the result of Borel and Serre.

This is based on joint work with Radhika Gupta (see arXiv:1810.09380).

#### Yuri Santos

*Soluble matrix groups and their finiteness lengths*

The finiteness length is a useful invariant which allows one to distinguish and describe groups to a certain degree. In this talk we recall the definition of the finiteness length and discuss the ongoing challenge of computing it for soluble linear groups, focusing on famous examples.

#### Olga Varghese

*Abstract group homomorphisms between locally compact groups and discrete groups*

We consider abstract group homomorphisms from l.c. groups into groups which are in the class

We show that any abstract homomorphism f from a locally compact group into a group H, where H is in the class G, is continuous or the image of f lies in the normalizer of a finite non-trivial subgroup of H. In particular, any abstract homomorphism from a locally compact group into a right angled Artin group is continuous.

This is joint work with L. Kramer.

#### Nils Leder

*Property FA for Automorphism Groups of small Graph Products*

We discuss the question whether the automorphism group Aut(G) of a graph product of finite cyclic groups has Serre's property FA.

If we fix a prime power n=p^{k}, k ≥ 1 and assume all vertex groups to be cyclic of order n, we can give a full classification for small graph products, i.e. graph products whose defining graph has at most 5 vertices.

The main tools involved are results on special graph products (free products, circles) and some reduction techniques. The latter use characteristic subgroups to reduce the question for a given graph product G to a graph product G' with a smaller number of vertices.