AEI-DFG: Representation zeta functions of arithmetic, compact analytic and branch groups

• Applicants: Dr. Steffen Kionke; Professor Dr. Benjamin Klopsch
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 541392601

An important discipline within group theory is concerned with the study of linear representations of groups, i.e., realizations of groups and their quotients as matrix groups over a field, such as the complex numbers or different, perhaps finite fields. Even for special classes of groups, whose structure may be comparatively well understood, it is difficult to work out all possible representations. Often it is not even possible - in a suitable technical sense - to give an overview over all irreducible representations. Using tools from number theory, in particular Dirichlet generating functions, we are able to encode the asymptotic distribution of representations for such groups. Subsequently we can apply fresh techniques - for instance, from geometry - to explore features of this distribution. The resulting representation zeta functions constitute far reaching generalizations of the famous Riemann zeta function. In the recent past interesting advances have be made in the study of zeta functions capturing information about complex representations of arithmetic groups and p-adic Lie groups. Recent studies into probabilistic properties of groups have thrust representations over finite fields into the limelight. The aim of the proposed project is to advance the study of representation zeta functions associated to three large classes of groups, in a concerted approach: (i) arithmetic groups, (ii) compact non-archimedean Lie groups and (iii) profinite branch groups. The latter form a special class of groups of automorphisms of infinite rooted trees; on the one hand they are connected to the non-archimedean Lie groups, on the other hand they feature a rather different internal algebraic structure. The conventional approach has been to focus almost exclusively on complex representations. In contrast, we take a keen interest in representations defined over basic fields in number theory, such as finite fields and algebraic number fields. This extends the scope of the subject in several ways. A significantly larger class of groups can be investigated, and at the same time new arithmetic phenomena arise in connection with the choice of ground field. We expect to develop strategies and methods that will have relevant applications also in other contexts. The project incorporates concrete milestones which form part of a long term development of asymptotic group theory.

• DFG Programme: Research Grants
• International Connection: Spain
• Partner Organisation: Agencia Estatal de Investigación
• Cooperation Partner: Privatdozent Dr. Jon Gonzalez-Sanchez