$H^2(SL_3(\mathbb{Z}[t]); \mathbb{Q})$ is infinite dimensional

TL;DR

This paper demonstrates that the second rational cohomology group of the special linear group over polynomial integers in three variables is infinite dimensional, using geometric and algebraic tools related to Euclidean buildings and Morse theory.

Contribution

It establishes the infinite dimensionality of $H^2(SL_3(bZ[t]); bQ)$, extending recent methods to this specific algebraic group setting.

Findings

$H^2(SL_3(bZ[t]); bQ)$ is infinite dimensional

Uses Euclidean building for $SL_3(bQ((t^{-1})))$

Employs Morse function from Bux-Köhl-Witzel

Abstract

We prove that $H^2(SL_3(\mathbb{Z}[t]); \mathbb{Q})$ is infinite dimensional. The proof follows an outline similar to recent results by Cobb, Kelly, and Wortman, using the Euclidean building for $SL_3(\mathbb{Q}((t^{-1})))$ and a Morse function from Bux-K\"ohl-Witzel.