Statistics of subgroups of the modular group
| Autor: | Bassino, Frédérique, Nicaud, Cyril, Weil, Pascal |
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| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | International Journal of Algebra and Computation, 31:08 (2021), pages 1691-1751 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1142/S0218196721500624 |
| Popis: | 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 $\Gamma(H)$, we consider the number of vertices of $\Gamma(H)$ 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 $\textsf{PSL}(2,\mathbb{Z})$. Comment: 62 pages. Typos fixed. Lemma 2.8, which was not correct as stated, has been reworked |
| Databáze: | arXiv |
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