On homotopy invariance for homology of rank two groups

We show that 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, we obtain examples of rings where the Steinberg group St3 is not a central extension of the elementary group E3. We also show that homotopy invariance works for the Steinberg groups of rank two groups over integral domains with many units.