On finite and elementary generation of SL₂(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} $\Gamma$ of $SL_2(F)$ containing $SL_2(R)$? 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 $SL_2(R)$ 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 $GE_2$ property introduced by P.M. Cohn.