Amalgamated free products, unstable homotopy invariance, and the homology of SL2(Z[t])
Abstract
We show that if R is an integral domain with many units, then the inclusion E2(R) --> E2(R[t]) induces an isomorphism in integral homology. This is a consequence of the existence of an amalgamated free product decomposition for E2(R[t]). We also use this decomposition to study the homology of E2(Z[t]). We show that Hi(E2(Z[t]),Z) contains a countable rank free summand for each i>0 and that this summand maps nontrivially into Hi(SL2(Z[t]),Z); hence, the latter is not finitely generated. This improves on a result of Grunewald, et.al., which states that SL2(Z[t]) has free quotients of countable rank (and hence, H1(SL2(Z[t]) is not finitely generated).
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