Robin J. Sroka: Steinberg modules and (co-)homology of arithmetic groups

Abstract.

While Borel computed the rational cohomology of \(\mathrm{GL}_n(\mathbb{Z})\), \(\mathrm{SL}_n(\mathbb{Z})\) and \(\mathrm{Sp}_{2n}(\mathbb{Z})\) in degrees that are small compared to n, little is known about it in high cohomological degrees.

Part 1: Vanishing patterns in high degrees

In the first part, I will discuss recent work exploring the rational cohomology in high degrees with a focus on \(\mathrm{Sp}_{2n}(\mathbb{Z})\). The starting point is a duality theorem of Borel–Serre, which shows that the rational cohomology of \(\mathrm{Sp}_{2n}(\mathbb{Z})\) is trivial in degrees greater than \(n^2\) and that, in degree \(n^2\) and below, it is controlled by a complicated \(\mathrm{Sp}_{2n}(\mathbb{Z})\)-representation: the symplectic Steinberg module. I will explain how a suitable generating set and presentation of this module can be used to calculate the rational cohomology of \(\mathrm{Sp}_{2n}(\mathbb{Z})\) in degree \(n^2\) and \(n^2 - 1\), discuss connections to the study of moduli spaces, a conjecture of Church–Farb–Putman as well as guiding analogies with \(\mathrm{SL}_n(\mathbb{Z})\). This is based on joint works with Brück–Miller–Patzt–Wilson, Brück–Patzt and Brück–Santos Rego.

Part 2: Stability patterns in small degrees

A tool that, in many cases, makes rational (co-)homology computations in small degrees possible is (co-)homological stability. In the second part, I will discuss work in progress with Bernard–Miller in which we establish a generic slope-1 stability result for the rational homology of general linear groups \(\mathrm{GL}_n(R)\) of a large class of rings \(R\), including all Euclidean rings. This extends classical slope-1/2 stability results due to Charney, van der Kallen and Maazen, and builds on recent ideas of Galatius–Kupers–Randal-Williams and Kupers–Miller–Patzt. Our proof relies on a new method for constructing generating sets of relative Steinberg modules.