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On the groups $\mathrm{SL}\sb 2(\mathbb{Z}[x])$ and $\mathrm{SL}\sb 2(k[x,y])$. (English) Zbl 0805.20042
Let $R = \bbfZ[x]$ or $k[x,y]$, where $\bbfZ$ is the ring of rational integers and $k$ is a finite field, and let $U\sb 2(R)$ be the subgroup of $\text{SL}\sb 2(R)$ generated by the unipotent matrices. The authors prove that $\text{SL}\sb 2(R)/U\sb 2(R)$ has a free quotient of any finite rank. This result is very much a two-dimensional anomaly since Suslin has proved that, when $n \geq 3$, $\text{SL}\sb n(R) = E\sb n(R)$, where $E\sb n(R)$ 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 $\phi: \text{SL}\sb 2(R) \to \text{SL}\sb 2(S)$ is “almost surjective”. The proof of the main result follows from looking at $\text{Im }\phi$ for various $S$. When $R = \bbfZ[x]$, $S$ 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 $U\sb 2(R)/\widehat{E}\sb 2(R)$, when $\widehat{E}\sb 2(R)$ is the normal subgroup of $SL\sb 2(R)$ generated by $E\sb 2(R)$.

20H25Other matrix groups over rings
20H10Fuchsian groups and their generalizations (group theory)
20E07Subgroup theorems; subgroup growth
20E05Free nonabelian groups
20F05Generators, relations, and presentations of groups
19B14Stability for linear groups ($K_1$)
11R58Arithmetic theory of algebraic function fields
20G35Linear algebraic groups over adèles and other rings and schemes
Full Text: DOI
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