The integral group ring ${\mathbb Z}G$ of a finite group is a ring of fundamental interest that gives an obvious link between group and ring theory. In this context, its group of invertible elements ${\mathcal U}({\mathbb Z} G)$ is an object of crucial importance. It remains a challenge to refine our understanding of the structure of this finitely presented group. Because ${\mathbb Z}G$ is an order in the rational group algebra, i.e. it is a finitely generated $\mathbb{Z}$-module that contains a $\mathbb{Q}$-basis of the finite dimensional $\mathbb{Q}$-algebra $\mathbb{Q} G$, it is natural to consider the problems in the wider context of unit groups of orders.

The aim of this lecture is to give, more or less, an up to date state of the art of some of the knowledge of the unit group ${\mathcal U}({\mathbb Z} G)$ and some of the important problems.

We will mainly focus on the following issues: (1) constructions of units, (2) constructions of large central subgroups, (3) constructions of subgroups of finite index, (4) rational representations of $G$, (5) constructions of idempotents, (6) open problems and a link with discontinuous actions on hyperbolic spaces.

When? | 04.11.2014 17:15 |
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Where? | PER 08 Phys 2.52 Chemin du Musée 3 1700 Fribourg |

Contact | Department of Mathematics isabella.schmutz@unifr.ch |