# ALGB Seminar: Doryan Temmerman

Title: A study of property (FA) for low-rank special linear groups.

Abstract:

When trying to study the $\mathcal{U}(\mathbb{Z}G)$, the unit group of the integral group ring of some finite group $G$, it is interesting to consider decompositions as amalgamated products. This leads us to studying Serre's property (FA).

It appears that we can reduce (modulo some obstacles) the question of "Does $\mathcal{U}(\mathbb{Z}G)$ have (FA)?" to the same question for special linear groups over orders, which are related to the Wedderburn-Artin decomposition of $\mathbb{Q}G$.

It is known that many of these special linear groups satisfy this property (FA), and even the stronger Kazhdan's property (T).

In this talk we will discuss those other cases (called exceptional of type (II)) and provide a proof of the fact that they do not satisfy (FA). This will lead us to discuss elementary linear groups, their presentations, abelianizations and amalgamated decompositions.

This is joint work with A. Bächle, G. Janssens, E. Jespers and A. Kiefer.