Bielefeld-Münster Seminar 06/15

Bielefeld-Münster Seminar

on Groups, Geometry and Topology

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.