{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,7]],"date-time":"2025-05-07T04:19:55Z","timestamp":1746591595026},"reference-count":37,"publisher":"World Scientific Pub Co Pte Ltd","issue":"08","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2021,12]]},"abstract":"<jats:p> 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]. <\/jats:p>","DOI":"10.1142\/s0218196721500624","type":"journal-article","created":{"date-parts":[[2021,10,12]],"date-time":"2021-10-12T01:21:31Z","timestamp":1634001691000},"page":"1691-1751","source":"Crossref","is-referenced-by-count":3,"title":["Statistics of subgroups of the modular group"],"prefix":"10.1142","volume":"31","author":[{"given":"Fr\u00e9d\u00e9rique","family":"Bassino","sequence":"first","affiliation":[{"name":"Universit\u00e9 Sorbonne Paris Nord, LIPN, CNRS UMR 7030, F-93430 Villetaneuse, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cyril","family":"Nicaud","sequence":"additional","affiliation":[{"name":"LIGM, Universit\u00e9 Gustave Eiffel, CNRS, ESIEE Paris, F-77454, Marne-la-Vall\u00e9e, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pascal","family":"Weil","sequence":"additional","affiliation":[{"name":"University of Bordeaux, CNRS, Bordeaux INP, LABRI, UMR 5800, F-33400 Talence, France"},{"name":"CNRS, ReLaX, UMI 2000, Siruseri, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2021,10,12]]},"reference":[{"key":"S0218196721500624BIB001","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20407"},{"key":"S0218196721500624BIB002","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196708004482"},{"key":"S0218196721500624BIB003","doi-asserted-by":"publisher","DOI":"10.1137\/1016082"},{"key":"S0218196721500624BIB004","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-9.3.451"},{"key":"S0218196721500624BIB005","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(98)00225-4"},{"key":"S0218196721500624BIB006","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1951-038-3"},{"key":"S0218196721500624BIB008","doi-asserted-by":"publisher","DOI":"10.1017\/S204061850003522X"},{"key":"S0218196721500624BIB009","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511801655"},{"key":"S0218196721500624BIB010","doi-asserted-by":"publisher","DOI":"10.1007\/BF02095994"},{"key":"S0218196721500624BIB011","doi-asserted-by":"publisher","DOI":"10.2307\/2944334"},{"key":"S0218196721500624BIB012","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1949-017-2"},{"key":"S0218196721500624BIB013","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.2001.9033"},{"key":"S0218196721500624BIB014","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2017.05.037"},{"key":"S0218196721500624BIB015","series-title":"Ausgearbeitet und vervollst\u00e4ndigt von Robert Fricke. Nachdruck der ersten Auflage. Bibliotheca Mathematica Teubneriana","volume-title":"Vorlesungen \u00fcber die Theorie der elliptischen Modulfunktionen. Band I Grundlegung der Theorie","author":"Klein F.","year":"1966"},{"key":"S0218196721500624BIB016","series-title":"Ausgearbeitet und vervollst\u00e4ndigt von Robert Fricke. Nachdruck der ersten Auflage. Bibliotheca Mathematica Teubneriana","volume-title":"Vorlesungen \u00fcber die Theorie der elliptischen Modulfunktionen. Band II Fortbildung und Anwendung der Theorie","author":"Klein F.","year":"1966"},{"key":"S0218196721500624BIB017","volume-title":"The Art of Computer Programming Volume\u00a0II Seminumerical Algorithms","author":"Knuth D. E.","year":"1969"},{"key":"S0218196721500624BIB018","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-8693(03)00515-5"},{"key":"S0218196721500624BIB019","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-8965-0"},{"key":"S0218196721500624BIB020","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196701000498"},{"key":"S0218196721500624BIB021","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196707003846"},{"issue":"3","key":"S0218196721500624BIB022","first-page":"529","volume":"244","author":"Mednyh A. D.","year":"1979","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"S0218196721500624BIB023","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-7643-8412-8_12"},{"key":"S0218196721500624BIB024","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-19.1.164"},{"key":"S0218196721500624BIB025","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-1.1.351"},{"key":"S0218196721500624BIB026","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1955-021-8"},{"key":"S0218196721500624BIB027","doi-asserted-by":"publisher","DOI":"10.1007\/s002220050091"},{"key":"S0218196721500624BIB028","doi-asserted-by":"publisher","DOI":"10.1112\/S0024610702003599"},{"key":"S0218196721500624BIB029","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2003.09.005"},{"key":"S0218196721500624BIB030","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2007.01.016"},{"key":"S0218196721500624BIB031","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2009.12.011"},{"key":"S0218196721500624BIB032","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1976-0466047-9"},{"key":"S0218196721500624BIB033","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196707004244"},{"key":"S0218196721500624BIB034","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4176-8"},{"key":"S0218196721500624BIB035","doi-asserted-by":"publisher","DOI":"10.1515\/jgth-2015-0028"},{"key":"S0218196721500624BIB036","doi-asserted-by":"publisher","DOI":"10.1007\/BF02095993"},{"key":"S0218196721500624BIB037","doi-asserted-by":"publisher","DOI":"10.1007\/BF01171151"},{"key":"S0218196721500624BIB038","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1978-0502015-8"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196721500624","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,29]],"date-time":"2021-12-29T06:45:46Z","timestamp":1640760346000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196721500624"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,12]]},"references-count":37,"journal-issue":{"issue":"08","published-print":{"date-parts":[[2021,12]]}},"alternative-id":["10.1142\/S0218196721500624"],"URL":"https:\/\/doi.org\/10.1142\/s0218196721500624","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,10,12]]}}}