有理Möbius群的自由性和$S$-算术性
Freeness and $S$-arithmeticity of rational Möbius groups
摘要：
我们提出了一个新的计算方法来解决一个经典问题：证明$\mathrm{SL}(2,\mathbb{Q})$的（$2$-生成器，抛物）莫比乌斯子群的非自由性。所使用的主要工具是Zagliki稠密群的算法和用于计算$\mathbb{Z}$的本地化$R= \mathbb{Z}[\frac{1}{b}]$的$\mathrm{SL}(2, R)$的表示的算法。证明了一个Möbius子群$G$在相应的$\mathrm{SL}(2, R)$中具有有限指数，从而证明了它是不自由的.进一步的信息，$G$的结构得到，例如，我们计算的最小子群的有限指数在$\mathrm{SL}(2,R)$包含$G$。
英文摘要：
We initiate a new, computational approach to a classical problem: certifying non-freeness of ($2$-generator, parabolic) Möbius subgroups of $\mathrm{SL}(2,\mathbb{Q})$. The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of $\mathrm{SL}(2, R)$ for a localization $R= \mathbb{Z}[\frac{1}{b}]$ of $\mathbb{Z}$. We prove that a Möbius subgroup $G$ is not free by showing that it has finite index in the relevant $\mathrm{SL}(2, R)$. Further information about the structure of $G$ is obtained; for example, we compute the minimal subgroup of finite index in $\mathrm{SL}(2,R)$ that contains $G$.
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