On Kazhdan's property (T) and completeness of automorphism groups of Coxeter groups

• Applicant: Dr. Olga Varghese
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 438625854

This project is located in the area of geometric group theory. The basic principle of geometric group theory is to investigate algebraic properties of groups using geometric and topological methods. In this project I focus on groups which are defined via edge-labeled graphs, Coxeter groups, and their automorphism groups. One important property of a group is property (T). It was defined by Kazhdan for locally compact groups in terms of unitary representations. This property was reformulated in many different mathematical areas, in particular in geometric group theory. The first goal of this project is to show that the automorphism group of an infinite Coxeter group $Aut(W_\Gamma)$ does not have property (T) and give algebraic reasons that prohibit this group from having property (T). Many of the groups in geometric group theory are rigid, in the sense that their outer automorphism groups are finite. The second goal of this project is to characterize those automorphism groups of Coxeter groups in terms of $\Gamma$ which are complete (i.e. $Aut(Aut(W_\Gamma))=Inn(Aut(W_\Gamma)))$. Along the way I plan to investigate further properties of the automorphism group of a Coxeter group.

• DFG Programme: Research Grants