{"title":"随机幂零群，多环表示和丢番图问题","authors":"A. Garreta, A. Myasnikov, D. Ovchinnikov","doi":"10.1515/gcc-2017-0007","DOIUrl":null,"url":null,"abstract":"Abstract We introduce a model of random finitely generated, torsion-free, 2-step nilpotent groups (in short, τ 2 {\\tau_{2}} -groups). To do so, we show that these are precisely the groups with presentation of the form 〈 A , C ∣ [ a i , a j ] = ∏ t = 1 m c t λ t , i , j ( 1 ≤ i < j ≤ n ) , [ A , C ] = [ C , C ] = 1 〉 {\\langle A,C\\mid[a_{i},a_{j}]=\\prod_{t=1}^{m}c_{t}^{\\lambda_{t,i,j}}(1\\leq i<j% \\leq n),\\,[A,C]=[C,C]=1\\rangle} , where A = { a 1 , … , a n } {A=\\{a_{1},\\dots,a_{n}\\}} and C = { c 1 , … , c m } {C=\\{c_{1},\\dots,c_{m}\\}} . Hence, a random G can be selected by fixing A and C, and then randomly choosing integers λ t , i , j {\\lambda_{t,i,j}} , with | λ t , i , j | ≤ ℓ {|\\lambda_{t,i,j}|\\leq\\ell} for some ℓ {\\ell} . We prove that if m ≥ n - 1 ≥ 1 {m\\geq n-1\\geq 1} , then the following hold asymptotically almost surely as ℓ → ∞ {\\ell\\to\\infty} : the ring ℤ {\\mathbb{Z}} is e-definable in G, the Diophantine problem over G is undecidable, the maximal ring of scalars of G is ℤ {\\mathbb{Z}} , G is indecomposable as a direct product of non-abelian groups, and Z ( G ) = 〈 C 〉 {Z(G)=\\langle C\\rangle} . We further study when Z ( G ) ≤ Is ( G ′ ) {Z(G)\\leq\\operatorname{Is}(G^{\\prime})} . Finally, we introduce similar models of random polycyclic groups and random f.g. nilpotent groups of any nilpotency step, possibly with torsion. We quickly see, however, that the latter yields finite groups a.a.s.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"129 1","pages":"115 - 99"},"PeriodicalIF":0.1000,"publicationDate":"2016-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"13","resultStr":"{\"title\":\"Random nilpotent groups, polycyclic presentations, and Diophantine problems\",\"authors\":\"A. Garreta, A. Myasnikov, D. Ovchinnikov\",\"doi\":\"10.1515/gcc-2017-0007\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract We introduce a model of random finitely generated, torsion-free, 2-step nilpotent groups (in short, τ 2 {\\\\tau_{2}} -groups). To do so, we show that these are precisely the groups with presentation of the form 〈 A , C ∣ [ a i , a j ] = ∏ t = 1 m c t λ t , i , j ( 1 ≤ i < j ≤ n ) , [ A , C ] = [ C , C ] = 1 〉 {\\\\langle A,C\\\\mid[a_{i},a_{j}]=\\\\prod_{t=1}^{m}c_{t}^{\\\\lambda_{t,i,j}}(1\\\\leq i<j% \\\\leq n),\\\\,[A,C]=[C,C]=1\\\\rangle} , where A = { a 1 , … , a n } {A=\\\\{a_{1},\\\\dots,a_{n}\\\\}} and C = { c 1 , … , c m } {C=\\\\{c_{1},\\\\dots,c_{m}\\\\}} . Hence, a random G can be selected by fixing A and C, and then randomly choosing integers λ t , i , j {\\\\lambda_{t,i,j}} , with | λ t , i , j | ≤ ℓ {|\\\\lambda_{t,i,j}|\\\\leq\\\\ell} for some ℓ {\\\\ell} . We prove that if m ≥ n - 1 ≥ 1 {m\\\\geq n-1\\\\geq 1} , then the following hold asymptotically almost surely as ℓ → ∞ {\\\\ell\\\\to\\\\infty} : the ring ℤ {\\\\mathbb{Z}} is e-definable in G, the Diophantine problem over G is undecidable, the maximal ring of scalars of G is ℤ {\\\\mathbb{Z}} , G is indecomposable as a direct product of non-abelian groups, and Z ( G ) = 〈 C 〉 {Z(G)=\\\\langle C\\\\rangle} . We further study when Z ( G ) ≤ Is ( G ′ ) {Z(G)\\\\leq\\\\operatorname{Is}(G^{\\\\prime})} . Finally, we introduce similar models of random polycyclic groups and random f.g. nilpotent groups of any nilpotency step, possibly with torsion. We quickly see, however, that the latter yields finite groups a.a.s.\",\"PeriodicalId\":41862,\"journal\":{\"name\":\"Groups Complexity Cryptology\",\"volume\":\"129 1\",\"pages\":\"115 - 99\"},\"PeriodicalIF\":0.1000,\"publicationDate\":\"2016-12-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"13\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Groups Complexity Cryptology\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1515/gcc-2017-0007\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Groups Complexity Cryptology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/gcc-2017-0007","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Random nilpotent groups, polycyclic presentations, and Diophantine problems
Abstract We introduce a model of random finitely generated, torsion-free, 2-step nilpotent groups (in short, τ 2 {\tau_{2}} -groups). To do so, we show that these are precisely the groups with presentation of the form 〈 A , C ∣ [ a i , a j ] = ∏ t = 1 m c t λ t , i , j ( 1 ≤ i < j ≤ n ) , [ A , C ] = [ C , C ] = 1 〉 {\langle A,C\mid[a_{i},a_{j}]=\prod_{t=1}^{m}c_{t}^{\lambda_{t,i,j}}(1\leq i
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