论文标题
模块化组亚组的统计数据
Statistics of subgroups of the modular group
论文作者
论文摘要
我们计算模块化组$ \ textsf {psl}（2，\ mathbb {z}）$的有限生成的子组。更准确地说：每个这样的子组$ h $都可以用其失速图$γ（h）$表示，我们认为$γ（h）$的顶点的数量为$ h $，我们计算了大小$ n $的子组。由于索引$ n $子组具有尺寸$ n $，因此我们的结果将$ \ textsf {psl} {psl}（2，\ mathbb {z}）$的有限索引子组的列举概括了已知结果。我们为$ \ textsf {psl}（2，\ mathbb {z}）$的有限生成的子组的数量以及有限索引子组，免费子组和自由有限有限索引子组的数量提供了渐近等效物。我们还给出了大小$ n $子组的同构类型的预期值，并证明了有关此值的大偏差声明。对于有限指数和免费亚组，证明了类似的结果。最后，我们展示了如何有效地在随机下均匀地生成$ \ textsf {psl}（2，\ mathbb {z}）$的大小$ n $ subgroup（有限索引子组，免费子组）。
We count the finitely generated subgroups of the modular group $\textsf{PSL}(2,\mathbb{Z})$. More precisely: each such subgroup $H$ can be represented by its Stallings graph $Γ(H)$, we consider the number of vertices of $Γ(H)$ to be the size of $H$ and we count the subgroups of size $n$. Since an index $n$ subgroup has size $n$, our results generalize the known results on the enumeration of the finite index subgroups of $\textsf{PSL}(2,\mathbb{Z})$. We give asymptotic equivalents for the number of finitely generated subgroups of $\textsf{PSL}(2,\mathbb{Z})$, as well as of the number of finite index subgroups, free subgroups and free finite index subgroups. We also give the expected value of the isomorphism type of a size $n$ subgroup and prove a large deviations statement concerning this value. Similar results are proved for finite index and for free subgroups. Finally, we show how to efficiently generate uniformly at random a size $n$ subgroup (resp. finite index subgroup, free subgroup) of $\textsf{PSL}(2,\mathbb{Z})$.
