On the groups \(\mathrm{SL}_ 2(\mathbb{Z}[x])\) and \(\mathrm{SL}_ 2(k[x,y])\)
Let \(R = \mathbb Z[x]\) or \(k[x,y]\), where \(\mathbb Z\) is the ring of rational integers and \(k\) is a finite field, and let \(U_ 2(R)\) be the subgroup of \(\text{SL}_ 2(R)\) generated by the unipotent matrices. The authors prove that \(\text{SL}_ 2(R)/U_ 2(R)\) has a free quotient of any finite rank. This result is very much a two-dimensional anomaly since Suslin has proved that, when \(n \geq 3\), \(\text{SL}_ n(R) = E_ n(R)\), where \(E_ n(R)\) is the subgroup generated by the elementary matrices. Using a technique involving symplectic matrices it is shown that, if \(S\) is a quotient of \(R\), then the corresponding natural map \(\phi: \text{SL}_ 2(R) \to \text{SL}_ 2(S)\) is ``almost surjective''. The proof of the main result follows from looking at \(\text{Im }\phi\) for various \(S\). When \(R = \mathbb Z[x]\), \(S\) is the ring of integers of some imaginary quadratic number fields. When \(R = k[x,y]\), \(S\) is a ring of integers in a hyperelliptic function field over \(k\). The paper also contains results on the exact stable range of various polynomial rings, together with structure theorems for the group \(U_ 2(R)/\widehat{E}_ 2(R)\), when \(\widehat{E}_ 2(R)\) is the normal subgroup of \(\text{SL}_ 2(R)\) generated by \(E_ 2(R)\).
scientific article; zbMATH DE number 590937
scientific article