On finite and elementary generation of SL_2(R)

Motivated by a question of A. Rapinchuk concerning general reductive groups, we are investigating the following question: Given a finitely generated integral domain $R$ with field of fractions $F$, is there a \emph{finitely generated subgroup} $\mathrm{\Gamma }$ of $S{L}_2(F)$ containing $S{L}_2(R)$? We shall show in this paper that the answer to this question is negative for any polynomial ring $R$ of the form $R=R_0[s,t]$, where $R_0$ is a finitely generated integral domain with infinitely many (non–associate) prime elements. The proof applies Bass–Serre theory and reduces to analyzing which elements of $S{L}_2(R)$ can be generated by elementary matrices with entries in a given finitely generated $R$–subalgbra of $F$. Using Bass–Serre theory, we can also exhibit new classes of rings which do not have the $G{E}_2$ property introduced by P.M. Cohn.