Homology and finiteness properties of SL_2(Z[t,t^{-1}])

TL;DR

This paper investigates the homology and finiteness properties of the group SL_2(Z[t,t^{-1}]), demonstrating that its second homology group is not finitely generated, which addresses an open question in the field.

Contribution

It proves that the second homology group of SL_2(Z[t,t^{-1}]) is not finitely generated, providing new insights into the group's algebraic structure.

Findings

H_2(SL_2(Z[t,t^{-1}]); Z) is not finitely generated

Answers an open question by Bux and Wortman

Advances understanding of homology in algebraic groups

Abstract

We show that the group $H_2(\slzti;\zz)$ is not finitely generated, answering a question mentioned by Bux and Wortman in \cite{bux}.