Author:  Kevin P. Knudson, Northwestern University
Title:  Amalgamated free products, unstable homotopy invariance, and
        the homology of SL_2(Z[t])

<p>We prove that if R is an integral domain with many units, then the
inclusion E<sub>2</sub>(R) ---> E<sub>2</sub>(R[t]) induces an
isomorphism in integral homology.  This is a consequence of the
existence of an amalgamated free product decomposition for
E<sub>2</sub>(R[t]).  We also use this decomposition to study the
homology of E<sub>2</sub>(<b>Z</b>[t]).  In particular,we show that the group
H<sub>i</sub>(E<sub>2</sub>(<b>Z</b>[t]),<b>Z</b>) contains a
countably generated free summand for each i.  We show that this
summand maps nontrivially into
H<sub>i</sub>(SL<sub>2</sub>(<b>Z</b>[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<sub>2</sub>(<b>Z</b>[t]) has countable free quotients (and hence
that H<sub>1</sub>(SL<sub>2</sub>(<b>Z</b>[t])) is not finitely
generated). 
</p>