Abstract: In this talk I will describe a project, joint with Peter Patzt, Jeremy Miller, and Dan Yasaki, concerning the top-degree cohomology of SL_n(O), where O is a number ring. I will explain the statement and ramifications of our main result: assuming the generalized Riemann hypothesis, the Steinberg module of SL_n(O) is generated by integral apartments if and only if the ring O is Euclidean. We also construct new cohomology classes in the top cohomology group of the special linear groups of some quadratic imaginary number rings. The key to our results is to study the topology of certain posets associated to the groups SL_n(O), which I will describe.