**Bielefeld-Münster Seminar**

Date: 12 June 2015

Venue: University of Münster, Einsteinstraße 62

All talks will take place in SR 0. Tea will be served in the same room during the breaks.

### Schedule

9:45 Uhr | Nils Leder |

The Stability Pair Spectral Sequence | |

10:45 Uhr | Coffee break |

11:00 Uhr | Jeroen Schillewaert |

Small maximal independent sets | |

12:00 Uhr | Lunch break |

13:30 Uhr | Ann Kiefer |

Generators for discrete subgroups of 2-by-2 matrices over rational Clifford groups | |

14:30 Uhr | Coffee break |

15:00 Uhr | Yuri Santos Rego |

The Golod-Shafarevich Inequality for some pro-p Groups |

### Abstracts

#### Nils Leder

*The Stability Pair Spectral Sequence*

The stability pair spectral sequence was one main tool in my Master thesis where I studied homological stability for families of spherical Coxeter groups. Briefly, this spectral sequence describes the group homology of a Coxeter group G in terms of certain parabolic subgroups.

First, we will introduce the definition of spectral sequences and the notion of convergence. Then, we consider a filtration of the Coxeter complex by types. We use this to obtain an exact sequence of G-modules from which the stability pair spectral sequence is derived.

#### Jeroen Schillewaert

*Small maximal independent sets*

We call a d-regular graph δ-sparse if the number of paths of length two joining any pair of vertices is at most d^{1−δ}. Our main theorem shows that δ-sparse graphs have small maximal independent sets. This theorem has applications to a range of problems in finite geometry.

#### Ann Kiefer

*Generators for discrete subgroups of 2-by-2 matrices over rational Clifford groups*

In [1], we developed an algorithm to determine generators for discrete subgroups of quaternion algebras over quadratic imaginary extensions of ℚ or discrete subgroups of 2-by-2 matrices over quadratic imaginary extensions of ℚ. These groups act discontinuously on hyperbolic 3-space. The mentioned algorithm constructs a polyhedron containing a fundamental domain in order to find a set of generators.

In this work we adapt this algorithm to Clifford matrices acting discontinuously on hyperbolic n-space. The motivation behind it is to get a set of generators for discrete subgroups of 2-by-2 matrices over rational quaternion algebras

[1] E. Jespers, S. O. Juriaans, A. Kiefer, A. de A. e Silva, and A. C. Souza Filho. From the Poincaré theorem to generators of the unit group of integral group rings of finite groups. *Math. Comp.*, 84(203):1489-1520, 2015.

#### Yuri Santos Rego

*The Golod-Shafarevich Inequality for some pro-p Groups*

In our Master thesis we studied the main results given by J. Wilson in his paper "Finite presentations of pro-p groups and discrete groups", which extend the Golod-Shafarevich Inequality to some large classes of infinite pro-p groups. We will give some definitions and basic properties from the pro-p theory and relate the free (finitely generated) pro-p group to completed group algebras.

Next, we explore such relationship to obtain the main theorem, extracting some structural implications on groups for which the Inequality holds. Some applications for abstract groups include a version of the Inequality for those groups and Golod's construction for the General Burnside Problem.