**Bielefeld-Münster Seminar**

Date: 2 Feburary 2018

Venue: Bielefeld University

The morning talks will take place in U5-133, the afternoon talks in V2-213. Coffee will be served in the same rooms during the break.

### Schedule

10:00 Uhr | Yuri Santos |

Thompson, conjugacy and braids | |

11:00 Uhr | Coffee break |

11:30 Uhr | Stefan Witzel |

Boundary braids | |

12:30 Uhr | Lunch break |

14:00 Uhr | Nils Leder |

Property FA for Automorphism Groups of Graph Products | |

15:00 Uhr | Coffee break |

15:30 Uhr | Olga Varghese |

Linear representations of groups with property FA_{d} | |

16:30 Uhr | End the of seminar |

### Abstracts

#### Yuri Santos

*Thompson, conjugacy and braids*

I will report on developments and work in progress on decision problems for Thompson groups.

We will introduce Thompson's groups F and V and recall the state of knowledge of Dehn's fundamental problems for those groups. We then switch to braid groups and recall how one of those fundamental problems, namely the conjugacy problem, translates to a topological statement. Going back to Thompson, we shall see how the work of Belk and Matucci yields a similar topological interpretation for the conjugacy problem for F and V. Then, we present a strategy to solve the conjugacy problem for V_{br}, the braided variant of V, which reformulates the problem into a question about recognition of certain 3-manifolds.

#### Stefan Witzel

*Boundary braids*

I will present joint work with Michael Dougherty and Jon McCammond on so-called boundary braids. A boundary braid in a solid cylinder is defined by the condition that certain strands stay in the boundary cylinder.

Our motivation to study boundary braids is to better understand Brady-McCammond's dual braid complex. We show that the subcomplex of boundary braids decomposes as a metric direct product into the subcomplex associated to a parabolic subgroup and a euclidean polyhedron.

#### Nils Leder

*Property FA for Automorphism Groups of Graph Products*

Currently, I am interested in the following question:

Let G=W(Γ, G_{Γ}) be a graph product of finite cyclic groups. Does Aut(G) satisfy the fixed point property FA?

In this talk, we look at this from two perspectives: On the one hand, we show that if the defining graph Γ is a circle (and all vertex groups are of the same order), then Aut(G) has FA.

On the other hand, we study a reduction procedure involving characteristic subgroups which can often be used to prove that Aut(G) does __not__ satisfy FA.

Given a characteristic subgroup H of the graph product G, we can consider the natural homomorphism Aut(G) → Aut(G/H). In many cases, Aut(G/H) is an "obstruction" for Aut(G) in the sense that if Aut(G) has property FA, Aut(G/H) must satisfy FA as well.

As examples, we take the characteristic subgroup H to be the center Z=Z(G) or a subgroup defined by certain properties of the defining graph Γ.

#### Olga Varghese

*Linear representations of groups with property FA*

_{d}We show that (d+1)-dimensional linear representations over fields in positive characteristic of groups having property FA_{d} have finite image.