TY - JOUR
T1 - Metabelian groups
T2 - Full-rank presentations, randomness and Diophantine problems
AU - Garreta, Albert
AU - Legarreta, Leire
AU - Miasnikov, Alexei
AU - Ovchinnikov, Denis
N1 - Publisher Copyright:
© 2020 Walter de Gruyter GmbH, Berlin/Boston 2021Russian Science Foundation19-11-00209European Research CouncilPCG-336983Eusko JaurlaritzaIT974-16Ministerio de Economía, Industria y Competitividad, Gobierno de EspañaMTM2017-86802-PThis work was supported by the Russian Science Foundation grant project 19-11-00209. Additionally, the first named author was supported by the ERC grant PCG-336983. The first and second named authors were supported by the Basque Government grant IT974-16, and by the Ministry of Economy, Industry and Competitiveness of the Spanish Government Grant MTM2017-86802-P.
PY - 2021/5/1
Y1 - 2021/5/1
N2 - We study metabelian groups G given by full rank finite presentations 〈A|-|R〉M in the variety M of metabelian groups. We prove that G is a product of a free metabelian subgroup of rank max {0,|A|-|R|} and a virtually abelian normal subgroup, and that if |R|≤|A|-2, then the Diophantine problem of G is undecidable, while it is decidable if |R|≥|A|. We further prove that if |R|≤|A|-1, then, in any direct decomposition of G, all factors, except one, are virtually abelian. Since finite presentations have full rank asymptotically almost surely, metabelian groups finitely presented in the variety of metabelian groups satisfy all the aforementioned properties asymptotically almost surely.
AB - We study metabelian groups G given by full rank finite presentations 〈A|-|R〉M in the variety M of metabelian groups. We prove that G is a product of a free metabelian subgroup of rank max {0,|A|-|R|} and a virtually abelian normal subgroup, and that if |R|≤|A|-2, then the Diophantine problem of G is undecidable, while it is decidable if |R|≥|A|. We further prove that if |R|≤|A|-1, then, in any direct decomposition of G, all factors, except one, are virtually abelian. Since finite presentations have full rank asymptotically almost surely, metabelian groups finitely presented in the variety of metabelian groups satisfy all the aforementioned properties asymptotically almost surely.
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U2 - 10.1515/jgth-2020-0091
DO - 10.1515/jgth-2020-0091
M3 - Article
AN - SCOPUS:85097895466
SN - 1433-5883
VL - 24
SP - 453
EP - 466
JO - Journal of Group Theory
JF - Journal of Group Theory
IS - 3
ER -