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

TL;DR

This paper demonstrates that a certain subgroup of SL_3 over polynomial integers has an infinite-dimensional second cohomology group, using geometric methods related to Euclidean buildings.

Contribution

It establishes the existence of a finite index subgroup with infinite-dimensional cohomology in SL_3 over polynomial rings, a novel result in the field.

Findings

Existence of a finite index subgroup with infinite-dimensional H^2.

Use of Euclidean building geometry in cohomology analysis.

Advancement in understanding the cohomological properties of linear groups.

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 $\textbf{SL}_3(\mathbb{Q}((t^{-1})))$.