Amalgamated free products, unstable homotopy invariance, and the homology of SL_2(Z[t]), by Kevin P. Knudson

We prove 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]). In particular,we show that the group Hi(E2(Z[t]),Z) contains a countably generated free summand for each i. We show that this summand maps nontrivially into Hi(SL2(Z[t])) and hence the latter group is not finitely generated for all i. This improves on a result of Grunewald, Mennicke, and Vaserstein which states that SL2(Z[t]) has countable free quotients (and hence that H1(SL2(Z[t])) is not finitely generated).


Kevin P. Knudson <knudson@math.nwu.edu>