An infinitely generated virtual cohomology group for noncocompact arithmetic groups over function fields

Abstract

Let $\mathbf{G}({\mathcal{O}}_S)$ be a noncocompact irreducible arithmetic group over a global function field $K$ of characteristic $p$, and let $\Gamma$ be a finite-index, residually $p$-finite subgroup of $\mathbf{G}({\mathcal{O}}_S)$. We show that the cohomology of $\Gamma$ in the dimension of its associated Euclidean building with coefficients in the field of $p$ elements is infinite.