Amalgamated free products, unstable homotopy invariance, and the homology of SL_2(Z[t])

Abstract

We show that if R is an integral domain with many units, then the inclusion E_2(R) --> E_2(R[t]) induces an isomorphism in integral homology. This is a consequence of the existence of an amalgamated free product decomposition for E_2(R[t]). We also use this decomposition to study the homology of E_2(Z[t]). We show that H_i(E_2(Z[t]),Z) contains a countable rank free summand for each i>0 and that this summand maps nontrivially into H_i(SL_2(Z[t]),Z); hence, the latter is not finitely generated. This improves on a result of Grunewald, et.al., which states that SL_2(Z[t]) has free quotients of countable rank (and hence, H_1(SL_2(Z[t]) is not finitely generated).