论文标题
FreeNESS和$ S $ - 理性Möbiusgroups
Freeness and $S$-arithmeticity of rational Möbius groups
论文作者
论文摘要
我们针对经典问题启动了一种新的计算方法：认证（$ 2 $ - 生成剂，抛物线）Möbius子组的$ \ mathrm {sl}（2，\ mathbb {q}）$。使用的主要工具是Zariski密集组的算法和算法，用于计算$ \ Mathrm {sl}（2，r）$的演示文稿，用于本地化$ r = \ Mathbb {Z} [\ frac {\ frac {1} {1} {B} {B} {B} {B}]我们证明，Möbius子组$ g $是通过证明其在相关$ \ Mathrm {SL}（2，r）$中具有有限索引来免费的。有关$ g $结构的更多信息；例如，我们计算包含$ g $的$ \ mathrm {sl}（2，r）$中的有限索引的最小亚组。
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 $G$.
