Groups Generated by Root Unipotents: Higher-rank and rank-one
Abstract
For n≥ 3, let Γ(Q)=⟨ Eij(qij):i≠ j⟩ be the subgroup of SL(n,ℝ) generated by elementary matrices with nonzero rational parameters qij. 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, SL(n,ℤ[1/N]), n≥3, and a broad class of groups SL(2,𝒪F,S) have arbitrarily small two-generated finite-index subgroups. We then study the rank-one family Γq=⟨1&1 0&1 , 1&0 q&1 ⟩, q=s/t∈ℚ. For q≠0,±3, we prove that Γq=Γ1(t)(s) if and only if its upper-triangular subgroup strictly contains ⟨1&1 0&1⟩. We then apply the rank-one criterion to constructions arising from non-freeness, verifying it for q=s/t∈(−4,4) with 1≤ |s|≤21, and obtaining infinite families from indefinite binary quadratic forms and Pell-type equations.
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BibTeX
@misc{hao2026groups,
title = {{Groups Generated by Root Unipotents: Higher-rank and rank-one}},
author = {Yanlong Hao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.04580},
eprint = {2607.04580},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04580},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Groups Generated by Root Unipotents: Higher-rank and rank-one
AU - Yanlong Hao
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2607.04580
UR - https://arxiv.org/abs/2607.04580
AB - For n≥ 3, let Γ(Q)=⟨ E_ij(q_ij):i≠ j⟩ be the subgroup of SL(n,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, SL(n,ℤ[1/N]), n≥3, and a broad class of groups SL(2,𝒪_F,S) have arbitrarily small two-generated finite-index subgroups. We then study the rank-one family Γ_q=⟨1&1 0&1 , 1&0 q&1 ⟩, q=s/t∈ℚ. For q≠0,±3, we prove that Γ_q=Γ_1^(t)(s) if and only if its upper-triangular subgroup strictly contains ⟨1&1 0&1⟩. We then apply the rank-one criterion to constructions arising from non-freeness, verifying it for q=s/t∈(−4,4) with 1≤ |s|≤21, and obtaining infinite families from indefinite binary quadratic forms and Pell-type equations.
ER -