High-dimensional cohomology of arithmetic groups

Basic data for this project

Description

Arithmetic groups are algebraic structures that occur in many areas of mathematics such as number theory, topology, representation theory and of course group theory. Originally, they were used to study quadratic equation, but they also describe symmetries of important geometric objects. Although they have been studied for a long time, many questions remain open. In particular, we still know surprisingly little about their cohomology, a concept that allows one to measure invariants of groups in different dimensions. For “small” arithmetic groups, that is groups of low rank, one can calculate these invariants using computers. With growing rank and in higher dimensions, these calculations however quickly become too demanding. Homological stability techniques offer a very helpful approach to handle this complexity. They allow one to calculate the cohomology of an infinite family of groups of growing rank at once. Unfortunately, these techniques classically only apply to low-dimensional cohomology. In contrast to that, this project will investigate the high-dimensional cohomology of arithmetic groups. Several exciting discoveries have been made in this area recently. At the moment, these are limited to only a few groups, in particular the special linear group and the symplectic group over the integers. Nonetheless, patterns start to arise from these results. This project will systematically investigate these patterns and their boundaries. It aims for results that are valid for many types of groups at once, namely for all Chevalley groups. This new approach will for the first time allow a conceptual understanding of high-dimensional cohomology in a broad context. An important technical ingredient will be a duality result of Borel–Serre. It allows one to compute high-dimensional cohomology via low-dimensional homology, which is in principle much more accessible. The downside is that this dimension reduction is only possible after a change to coefficients in the so-called Steinberg module. This is why a good understanding of this object is a main goal of this project. To obtain this understanding, we will use a combination of algebraic and geometric, topological methods. For the topological methods, polyhedral complexes will play a key role. On the algebraic side, we will investigate connections to the algebraic K-theory of rings of integers.