{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,3,23]],"date-time":"2024-03-23T04:11:53Z","timestamp":1711167113093},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":1472,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2010,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let<jats:italic>G<\/jats:italic>be a group definable in a monster model<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200003017_inline1\" \/>of a rosy theory satisfying NIP. Assume that<jats:italic>G<\/jats:italic>has hereditarily finitely satisfiable generics and 1 &lt;<jats:italic>U<\/jats:italic><jats:sup>b<\/jats:sup>(<jats:italic>G<\/jats:italic>) &lt; \u221e. We prove that if<jats:italic>G<\/jats:italic>acts definably on a definable set of<jats:italic>U<\/jats:italic><jats:sup>\u0440<\/jats:sup>-rank 1, then, under some general assumption about this action, there is an infinite field interpretable in<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200003017_inline1\" \/>. We conclude that if<jats:italic>G<\/jats:italic>is not solvable-by-finite and it acts faithfully and definably on a definable set of<jats:italic>U<\/jats:italic><jats:sup>\u0440<\/jats:sup>-rank 1, then there is an infinite field interpretable in<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200003017_inline1\" \/>. As an immediate consequence, we get that if<jats:italic>G<\/jats:italic>has a definable subgroup<jats:italic>H<\/jats:italic>such that<jats:italic>U<\/jats:italic><jats:sup>\u0440<\/jats:sup>(<jats:italic>G<\/jats:italic>) =<jats:italic>U<\/jats:italic><jats:sup>\u0440<\/jats:sup>(<jats:italic>H<\/jats:italic>) + 1 and<jats:italic>G<\/jats:italic>\/\u22c2<jats:sub><jats:italic>g<\/jats:italic>\u2208<jats:italic>G<\/jats:italic><\/jats:sub><jats:italic>H<jats:sup>g<\/jats:sup><\/jats:italic>is not solvable-by-finite, then an infinite field interpretable in<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200003017_inline1\" \/>also exists.<\/jats:p>","DOI":"10.2178\/jsl\/1264433927","type":"journal-article","created":{"date-parts":[[2010,1,25]],"date-time":"2010-01-25T15:38:59Z","timestamp":1264433939000},"page":"372-386","source":"Crossref","is-referenced-by-count":2,"title":["Fields interpretable in superrosy groups with NIP (the non-solvable case)"],"prefix":"10.1017","volume":"75","author":[{"given":"Krzysztof","family":"Krupi\u0144ski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200003017_ref009","first-page":"1","volume":"71","author":"Onshuus","year":"2006","journal-title":"Properties and consequences of thorn-independence"},{"key":"S0022481200003017_ref008","doi-asserted-by":"publisher","DOI":"10.1112\/S0024610700001629"},{"key":"S0022481200003017_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF01270468"},{"key":"S0022481200003017_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2007.09.004"},{"key":"S0022481200003017_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-96-01623-6"},{"key":"S0022481200003017_ref001","unstructured":"Adler H. , A geometric introduction to forking and thorn-forking, preprint."},{"key":"S0022481200003017_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-017-3002-0"},{"key":"S0022481200003017_ref005","first-page":"919","volume":"72","author":"Ealy","year":"2007","journal-title":"Characterizing rosy theories"},{"key":"S0022481200003017_ref010","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(99)00016-0"},{"key":"S0022481200003017_ref014","first-page":"98","article-title":"Uncountable categorical nilpotent groups and Lie algebras","volume":"26","author":"Zilber","year":"1982","journal-title":"Russian Mathematics (Izvestiya Vysshikh Uchebnykh Zavedeni\u01d0)"},{"key":"S0022481200003017_ref007","article-title":"Fields interpretable in rosy theories","author":"Krupi\u0144ski","journal-title":"Israel Journal of Mathematics"},{"key":"S0022481200003017_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/087"},{"key":"S0022481200003017_ref002","unstructured":"Adler H. , Thorn-forking as local forking, preprint."},{"key":"S0022481200003017_ref011","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198534372.001.0001","volume-title":"Geometric stability theory","author":"Pillay","year":"1996"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200003017","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,3,23]],"date-time":"2024-03-23T03:47:14Z","timestamp":1711165634000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200003017\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,3]]},"references-count":14,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2010,3]]}},"alternative-id":["S0022481200003017"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1264433927","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,3]]}}}