An Infinite Dimensional Virtual Cohomology Group of $\textbf{SL}_3(\mathbb{Z}[t])$

Abstract

We prove that $\textbf{SL}_3(\mathbb{Z}[t])$ has a finite index subgroup $\Gamma$ such that $H^2(\Gamma; \mathbb{Q})$ is infinite dimensional. The proof uses the geometry of the Euclidean building for $\textbf{SL}_3(\mathbb{Q}((t^{-1})))$.