On finite and elementary generation of SL_2(R)

TL;DR

This paper investigates the finite and elementary generation of SL_2 over certain rings, showing that for polynomial rings over finitely generated domains with infinitely many primes, no finitely generated subgroup contains SL_2(R).

Contribution

It proves the non-existence of finitely generated subgroups containing SL_2(R) for specific polynomial rings, extending understanding of elementary generation and the $GE_2$ property.

Findings

No finitely generated subgroup of SL_2(F) contains SL_2(R) for R = R_0[s,t] with R_0 having infinitely many primes.

New classes of rings without the $GE_2$ property are identified.

Application of Bass--Serre theory to analyze elementary matrices in this context.

Abstract

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.