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

Abstract

We study subgroups generated by prescribed unipotent elements. For $n\ge 3$, let \[ Γ(Q)=\langle E_{ij}(q_{ij}):i\neq j\rangle \] be the subgroup of $\mathrm{SL}(n,\mathbb{R})$ generated by elementary matrices with nonzero rational parameters $q_{ij}$. We prove that $\hat{I}“(Q)$ is always $S$-arithmetic, extending classical integral-parameter results to arbitrary rational parameters. Our method is effective: it determines the relevant ring of $S$-integers, a diagonal conjugating matrix, and an explicit description of the resulting subgroup by congruence conditions. 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 $q\ne 0,\pm 3$, we prove that \[ Γ_q=Γ_1^{(t)}(s) \] if and only if its upper-triangular subgroup strictly contains \[\left\langle\begin{pmatrix}1&1\\0&1\end{pmatrix}\right\rangle.\] Thus the congruence-subgroup problem is reduced to constructing a single upper-triangular element outside this cyclic subgroup. As applications, we reinterpret several constructions from the study of non-freeness as constructions of arithmetic groups. We verify the criterion for all rational parameters $q=\frac{s}{t}\in (-4,4)$ with $1\le |s|\le 21$, and obtain new infinite families of congruence subgroups from indefinite binary quadratic forms and Pell-type equations.