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(Fn) | |
| 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(Fn) and GLn(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 groupsI 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(Fn)The non-abelian finite simple group Ln(2) is a quotient of Aut(Fn) (factor out Fn' 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(Fn). 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 groupsThe matching complex Mmn of the complete bipartite graph Kmn 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 Sn is (c(n,n)+1)-generating for Sn.
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 Bn. We discuss this application as well as the connectivity of the arc matching complex, which appears to be lower than the connectivity of Mmn. For m ≤ n, we can estimate the connectivity from below by:
min(m, ⌊ n+1⁄2 ⌋) - 2
Benjamin Brück
Higher generating subgroups in Aut(Fn) and GLn(ℤ)Using the action of Aut(Fn) on the free factor complex, we identify highly generating families of "parabolic" subgroups in Aut(Fn). To prove higher generation, we generalise a result of Abels and Holz stating that the parabolic subgroups in GLn(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(Fn), 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.