In this talk I will give an overview of two related projects. The first project concerns the high-degree rational cohomology of the special linear group of a number ring $R$. Church--Farb--Putman conjectured that, when $R$ is the integers, these cohomology groups vanish in a range close to the virtual cohomological dimension. The groups $SL_n(R)$ satisfy a twisted analogue of Poincaré duality called virtual Bieri--Eckmann duality, and their rational cohomology groups are governed by $SL_n(R)$-representations called the Steinberg modules. I will discuss a recent proof of the "codimension two" case of the Church--Farb--Putman conjecture using the topology of certain simplicial complexes related to the Steinberg modules. The second project concerns Rognes’ connectivity conjecture on a family of simplicial complexes (the "common basis complexes") with implications for algebraic K-theory. I will describe work-in-progress proving Rognes' conjecture for fields, and its connections to $SL_n(R)$ and the Steinberg modules. This talk includes past and ongoing work joint with Benjamin Brück, Alexander Kupers, Jeremy Miller, Peter Patzt, Andrew Putman, and Robin Sroka.

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