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 finitely generated subgroup Γ of SL2(F ) containing SL2(R)? We shall show in this paper that the answer to this question is negative for any polynomial ring R of the form R = R0[s, t], where R0 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 SL2(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 GE2 property introduced by P.M. Cohn. Mathematics Subject Classification 2000: 20H05, 20E06, 20E08; 13F20.