Groups Generated by Root Unipotents: Higher-rank and rank-one

Abstract

For $n\geq 3$, let \[ Γ(Q)=\langle E_{ij}(q_{ij}):i\neq j\rangle \] be the subgroup of $\operatorname{SL}(n,\mathbb R)$ generated by elementary matrices with nonzero rational parameters $q_{ij}$. We prove that $Γ(Q)$ is always $S$-arithmetic, extending classical integral-parameter results to arbitrary rational parameters. Our method is effective. We also prove strong small-generation results for finite-index subgroups of $S$-arithmetic groups. In particular, $\operatorname{SL}\left(n,\mathbb Z\left[\frac1N\right]\right)$, $n\geq3$, and a broad class of groups $\operatorname{SL}(2,\mathcal O_{F,S})$ have arbitrarily small two-generated finite-index subgroups. We then study the rank-one family \[ Γ_q=\left\langle\begin{pmatrix}1&1\\0&1 \end{pmatrix}, \begin{pmatrix} 1&0\\q&1 \end{pmatrix}\right\rangle,\qquad q=\tfrac{s}{t}\in\mathbb Q. \] For $q\neq0,\pm3$, we prove that \[ Γ_q=Γ_1^{(t)}(s) \] if and only if its upper-triangular subgroup strictly contains \[\left\langle\begin{pmatrix}1&1\\0&1\end{pmatrix}\right\rangle.\] We then apply the rank-one criterion to constructions arising from non-freeness, verifying it for $q=\tfrac{s}{t}\in(-4,4)$ with $1\leq |s|\leq21$, and obtaining infinite families from indefinite binary quadratic forms and Pell-type equations.