Research

My research focuses on group rings and their unit groups. This is a very wide topic though and here are some areas I have contributed to:

Units in $\mathbb{Z}G$

A main problem here is to understand the finite subgroups of $U(\mathbb{Z}G)$ and how their structure is connected to the subgroups of $G$, which naturally embed in $\mathbb{Z}G$ and form the so-called trivial units. A theorem of G. Higman from 1940 shows that if $G$ is abelian, then any element of finite order in $U(\mathbb{Z}G)$ is trivial. Though this is not the case for non-abelian groups, one can still hope that units of finite order are ‘‘essentially’’ trivial. Of course this needs to be made more precise and the best one could hope for was conjectured by Zassenhaus:

Zassenhaus Conjecture: Any unit of finite order in $\mathbb{Z}G$ is trivial up to conjugation in $U(\mathbb{Q}G)$.

This was the first conjecture I studied during my PhD when I visited the University of Murcia together with Ángel del Río and Mauricio Caicedo. We were able to generalize a result of Hertweck to show that the conjecture holds for cyclic-by-abelian groups. After my PhD I came back to Murcia and we started tp deepen the ideas we had developed before which gave us new methods and new results, but also groups which were candidates for counterexamples. Finally, we showed together with Florian Eisele that these groups are counterexamples, which settled the conjecture. It would be very interesting