The University of Sydney - School of Mathematics and Statistics

On finite and elementary generation of $S{L}_2(R)$

Peter Abramenko (Virginia)

Abstract

Let $R$ be an integral domain which is finitely generated as a ring. Interesting questions regarding $S{L}_2(R)$ 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 $S{L}_2(F)$ containing $S{L}_2(R)$ is finitely generated. I will also explain why the action of $S{L}_2(F)$ on an appropriate (Bruhat-Tits) tree is an important ingredient of the proof of this theorem.