## On finite and elementary generation of \(SL_2(R)\)

Peter Abramenko (Virginia)

Abstract

Let \(R\) be an integral domain which is finitely generated as a ring. Interesting questions regarding \(SL_2(R)\) are whether this group is finitely generated or whether it is generated by elementary matrices. I will explain how these two questions are related, and present a brief survey of some well-known results in this context. The main part of the talk will be devoted to the following (new) Theorem: Let \(R_0\) be a finitely generated integral domain of Krull dimension greater than 1 (e.g. \(R_0 = Z[x]\) or \(F_q[x,y]),\) \(R = R_0[t]\) the polynomial ring over \(R_0\) and \(F\) the field of fractions of \(R.\) Then *no* subgroup of \(SL_2(F)\) containing \(SL_2(R)\) is finitely generated. I will also explain why the action of \(SL_2(F)\) on an appropriate (Bruhat-Tits) tree is an important ingredient of the proof of this theorem.