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

Abstract

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 $\text{SL}_n(\mathcal{O}/\mathfrak{p}).$ 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.