## 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.