High-degree cohomology of congruence subgroups of ${\text{SL}}_n(\mathcal{O})$ via cohomology of $S$-arithmetic groups

If $\mathfrak{p}$ is a prime ideal of a number ring $\mathcal{O}$, then the top-degree cohomology of the principal congruence subgroup of level $\mathfrak{p}$ is naturally a representation of ${\text{SL}}_n(\mathcal{O}\mathrm{/}\mathfrak{p})\mathrm{.}$ We prove that the multiplicity of the Steinberg representation in this cohomology space is one. When $\mathcal{O}$ is Euclidean and $\mathfrak{p}$ is suitably small – for example a universal side divisor – then we prove that the multiplicity of the Steinberg representation in the next-highest-degree cohomology space is zero. Our proof relies on a computation of the cohomology of an $S$-arithmetic group ouside of a linear range of degrees, derived from work of Blasius–Franke–Grunewald.