Unstable homotopy invariance and the homology of \(\text{SL}_2(\mathbb{Z}[t])\) (Q1568747)

We say that a ring \(R\) has many units if there are arbitrarily long finite sequences in \(R\) such that the sum of any nonempty subsequence is a unit. It is proved that if \(R\) is a commutative domain and has many units, then the natural inclusion \(E_2(R)\to E_2(R[t])\) induces an isomorphism in integral homology. This is a consequence of an amalgamated free product decomposition of \(E_2(R[t])\). This decomposition also exists for \(E_2(\mathbb{Z}[t])\) which is used to show that a great deal of homology of \(E_2(\mathbb{Z}[t])\) maps nontrivially into the homology of \(\text{SL}_2(\mathbb{Z}[t])\). As a consequence, it is shown that the latter is not finitely generated in all positive degrees.