Homotopy invariance for homology of linear groups: the case SL_4

TL;DR

This paper proves homotopy invariance of homology for SL_4 over certain rings by analyzing a contractible simplicial complex with CAT(0) properties derived from Bruhat-Tits buildings.

Contribution

It establishes homotopy invariance for the homology of SL_4 over local factorial rings using geometric group theory techniques.

Findings

The simplicial complex associated with E_4(R[t]) is contractible.

The complex satisfies the CAT(0) property for local factorial rings.

Homotopy invariance holds for the homology of SL_4 in this setting.

Abstract

In this paper, we investigate homotopy invariance for homology of SL_4. For any commutative ring, the group E_4(R[t]) acts on a simplicial complex whose contractibility implies homotopy invariance. We show that for a local factorial ring R, this complex satisfies the CAT(0)-property for the induced length metric from the Bruhat-Tits building.