### 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 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]). In particular,we show that the group
H_{i}(E_{2}(**Z**[t]),**Z**) contains a
countably generated free summand for each i. We show that this
summand maps nontrivially into
H_{i}(SL_{2}(**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
SL_{2}(**Z**[t]) has countable free quotients (and hence
that H_{1}(SL_{2}(**Z**[t])) is not finitely
generated).

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