由幂幺元生成的群：高阶与一阶

中文摘要 AI 辅助

我们研究由规定的幂幺元生成的子群。对于\(n\geq3\)，设\(\Gamma(Q)=\langle E_{ij}(q_{ij}):i\neq j\rangle\)是\(\operatorname{SL}(n,\mathbb R)\)中由具有非零有理参数\(q_{ij}\)的初等矩阵生成的子群。我们证明\(\Gamma(Q)\)总是\(S -\)算术的，将经典的整参数结果扩展到任意有理参数。我们的方法是有效的：它确定了相关的\(S -\)整数环、一个对角共轭矩阵以及通过同余条件对所得子群的明确描述。然后我们研究一阶族\(\Gamma_q = \left\langle \begin{pmatrix} 1&1\\ 0&1 \end{pmatrix}, \begin{pmatrix} 1&0\\ q&1 \end{pmatrix} \right\rangle\)，\(q=\tfrac{s}{t}\in\mathbb Q\)。对于\(q\neq0,\pm3\)，我们证明\(\Gamma_q=\Gamma_1^{(t)}(s)\)当且仅当其上三角子群严格包含\(\left\langle\begin{pmatrix}1&1\\0&1\end{pmatrix}\right\rangle\)。因此同余子群问题简化为在这个循环子群之外构造一个单一的上三角元素。作为应用，我们重新解释了从非自由性研究中的几个构造作为算术群的构造。我们验证了对于所有有理参数\(q=\tfrac{s}{t}\in(-4,4)\)且\(1\leq |s|\leq21\)的准则，并从不定二元二次型和佩尔型方程中获得了新的无限族同余子群。

英文摘要

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.

发表机构

• University of Michigan（密歇根大学）

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