On homotopy invariance for homology of rank two groups. (Q1934983)

Homotopy invariance fails for homology of elementary groups of rank two over integral domains which are not fields. The proof is an adaptation of the argument used by Behr to show that rank two groups are not finitely presentable. As a by-product, the author obtains examples of rings where the Steinberg group \(\text{St}_3\) is not a central extension of the elementary group \(E_3\). It is also shown that homotopy invariance works for the Steinberg groups of rank two groups over integral domains with many units.