Representation zeta functions associated to arithmetic groups and compact analytic groups

• Applicant: Professor Dr. Benjamin Klopsch
• Subject Area: Mathematics
• Term: from 2014 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 262827805

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. Even for special classes of groups, such as `semisimple' arithmetic groups, 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, certain Dirichlet generating functions, we are able to encode the asymptotic distribution of representations for such groups. Subsequently we can apply techniques from analysis, geometry and model theory to explore features of this distribution. The resulting representation zeta functions constitute farreaching generalizations of the famous Riemann zeta function. The latter plays a central role in number theory as it encodes information about the distribution of prime numbers. In recent years interesting advances have be made in the study of representation zeta functions of arithmetic groups and p-adic Lie groups. The aim of the project is to refine the original definition of a representation zeta function in different ways in order to put the emerging theory on a much wider basis. In one direction we will study representations over number fields instead of the complex numbers. In another direction we will associate a zeta function to each admissible infinite dimensional representation. For this purpose one needs to develop new methods that will have relevant applications also in other contexts. The project consists of concrete aims forming part of a long term strategy to extend asymptotic group theory.

• DFG Programme: Research Grants
• International Connection: Spain
• Participating Persons: Privatdozent Dr. Jon Gonzalez-Sanchez; Professor Dr. Andrei Jaikin Zapirain; Professor Dr. Christopher Voll