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

Abstract

Let ${\bf G} (\mathcal{O}\_S)$ 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 ${\bf G}(\mathcal{O}\_S)$. 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.