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Article Dans Une Revue International Journal of Algebra and Computation Année : 2021

Statistics of subgroups of the modular group

Résumé

We count the finitely generated subgroups of the modular group [Formula: see text]. More precisely, each such subgroup [Formula: see text] can be represented by its Stallings graph [Formula: see text], we consider the number of vertices of [Formula: see text] to be the size of [Formula: see text] and we count the subgroups of size [Formula: see text]. Since an index [Formula: see text] subgroup has size [Formula: see text], our results generalize the known results on the enumeration of the finite index subgroups of [Formula: see text]. We give asymptotic equivalents for the number of finitely generated subgroups of [Formula: see text], 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 [Formula: see text] subgroup and prove a large deviation 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 [Formula: see text] subgroup (respectively, finite index subgroup, free subgroup) of [Formula: see text].

Dates et versions

hal-03507020 , version 1 (03-01-2022)

Identifiants

Citer

Pascal Weil, Frédérique Bassino, Cyril Nicaud. Statistics of subgroups of the modular group. International Journal of Algebra and Computation, 2021, 31 (08), pp.1691-1751. ⟨10.1142/S0218196721500624⟩. ⟨hal-03507020⟩
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