**Bielefeld-Münster Seminar**

Date: 14 July 2017

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 | Cora Welsch |

The importance of maximal subgroups for the connectivity of the (finite index) coset poset of finitely generated infinite groups | |

11:00 | Coffee break |

11:30 | Barbara Baumeister |

The smallest non-abelian quotient of Aut(F_{n}) | |

12:30 | Lunch break |

14:00 | Kai-Uwe Bux |

Arc matching complexes and higher generation in braid groups | |

15:00 | Coffee break |

15:30 | Benjamin Brück |

Higher generating subgroups in Aut(F_{n}) and GL_{n}(ℤ) | |

16:30 | End the of seminar |

### Abstracts

#### Cora Welsch

*The importance of maximal subgroups for the connectivity of the (finite index) coset poset of finitely generated infinite groups*

I have proved that the nerve and the order complexes of the finite index coset poset, the poset of all cosets of all proper finite index subgroups, are contractible for many important classes of finitely generated infinite groups. Thus the upcoming question was: Are there groups such that the nerve and order complexes are not contractible? We can now answer this question in the affirmative and give examples. Moreover, I proved that also the coset poset, the poset of all cosets of all proper subgroups, of finitely generated infinite groups is sometimes contractible and sometimes not.

#### Barbara Baumeister

*The smallest non-abelian quotient of Aut(F*

_{n})The non-abelian finite simple group L_{n}(2) is a quotient of Aut(F_{n})
(factor out F_{n}' and then reduce modulo ℤ/2ℤ). In the talk
I will confirm the conjecture by Mecchia-Zimmermann that this is the
smallest non-abelian finite quotient of Aut(F_{n}). On the way some other
nice and new results will appear.

This is joint work with Dawid Kielak and Emilio Pierro.

#### Kai-Uwe Bux

*Arc matching complexes and higher generation in braid groups*

The matching complex M_{mn} of the complete bipartite
graph K_{mn} is c(m,n)-connected for:

c(m,n) = min(m, n, ⌊ ^{m+n+1}⁄_{3} ⌋) - 2

One can deduce that the family H of maximal subgroups in the
symmetric group S_{n} is (c(n,n)+1)-generating for S_{n}.

The arc matching complex for a disk with m marked points on the
boundary and n marked points in the interior yield a similar result
for higher generation in the braid group B_{n}. We discuss this
application as well as the connectivity of the arc matching complex,
which appears to be lower than the connectivity of M_{mn}.
For m ≤ n, we can estimate the connectivity from below by:

min(m, ⌊ ^{n+1}⁄_{2} ⌋) - 2

#### Benjamin Brück

*Higher generating subgroups in Aut(F*

_{n}) and GL_{n}(ℤ)Using the action of Aut(F_{n}) on the free factor complex, we identify highly generating families of "parabolic" subgroups in Aut(F_{n}). To prove higher generation, we generalise a result of Abels and Holz stating that the parabolic subgroups in GL_{n}(ℤ) are (n-2)-generating.
In order to achieve a uniform description of these phenomena, we exhibit more generally actions of groups on Cohen-Macaulay simplicial complexes and analyse the corresponding coset poset of the simplex stabilisers. Applying this to groups with BN-pairs and their actions on buildings, we recover the result of Abels and Holz as well as a similar statement about Levi-subgroups; in the case of Aut(F_{n}), the result mentioned above follows.

### Connections

There is an ICE connection leaving in Bielefeld at 8:22 and arriving in Münster at 9:22. There is also a local train connection leaving in Bielefeld at 7:58 and arriving in Münster at 9:17.

The following busses go from Münster station to Coesfelder Kreuz in the relevant period: line 5 from 9:22 to 9:36, line R64 from 9:27 to 9:40, line 34 from 9:30 to 9:46, line 13 from 9:31 to 9:48, line R63 from 9:43 to 9:54.