**Bielefeld-Münster Seminar**

Date: 24 June 2019

Venue: University of Münster, Einsteinstraße 62

The talks will be in room SR0 in the main building. Tea will be served in the same room during the breaks.

### Schedule

10:00 | Maneesh Thakur |

The inclusions D_{4}⊂ F_{4}⊂ E_{6} and a question of Albert | |

11:00 | Coffee break |

11:30 | Olga Varghese |

Graph products of finite groups and their automorphism groups | |

12:30 | Lunch break |

14:00 | Georges Neaime |

Interval Garside Structures for Euclidean Artin Groups | |

15:00 | Coffee break |

15:30 | Sophiane Yahiatene |

Hurwitz action in Coxeter groups and extended Weyl groups | |

16:30 | End the of seminar |

### Abstracts

#### Maneesh Thakur

*The inclusions D*

_{4}⊂ F_{4}⊂ E_{6}and a question of AlbertIn 1965 Albert raised the question if exceptional Jordan division algebras always contain cubic cyclic subfields. These algebras (now called Albert algebras) are of great importance for their connections with exceptional algebraic groups.

In our talk, we will explain the inclusions in the title and elaborate on the question of Albert in that context, also report on known results and some recent progress.

#### Olga Varghese

*Graph products of finite groups and their automorphism groups*

We investigate algebraic and geometric properties of groups whose building blocks are finite groups. These groups are defined via simplicial graphs Γ and are called graph products of finite groups G_{Γ}. We characterize many properties (e.g. word hyperbolicity, virtual freeness, Kazhdan's property (T)) of the graph product G_{Γ} and their automorphism group Aut(G_{Γ}) in terms of the shape of Γ.

#### Geoerges Neaime

*Interval Garside Structures for Euclidean Artin Groups*

Garside structures were invented in order to better understand Artin groups and their generalizations. Finite-type Artin groups admit two types of Garside structures corresponding to their standard and dual presentations. Concerning Euclidean Artin groups, Digne established Garside structures for two families of these groups by using their dual presentations. Recently, McCammond established that none of the remaining dual presentations (except for one additional case) correspond to Garside structures. He and Sulway also identified Euclidean Artin groups as subgroups of other Garside groups, thereby clarifying some of their properties. In this talk, shifting attention from dual presentations to other presentations for type Ã discovered by Shi and Corran-Lee-Lee, I will construct standard Garside structures for this type of Euclidean Artin groups.

#### Sophiane Yahiatene

*Hurwitz action in Coxeter groups and extended Weyl groups*

In the talk we will investigate a natural braid group action on factorizations of elements in reflection groups. First we will consider reflection factorizations of distinguished elements in Coxeter groups of finite rank and state conditions whether two factorizations lie in the same orbit under the natural action. Then we will define the so-called extended Weyl groups which can be seen as an extention of a family of Coxeter groups. They appear in the study of hereditary abelian categories. There we will consider the same action on reduced reflection factorizations of the class of Coxeter transformations. (j.w. B. Baumeister and P. Wegener)

The seminar is funded by the Cluster of Excellence "Mathematics Münster".