TY - JOUR
T1 - Full rank presentations and nilpotent groups
T2 - Structure, Diophantine problem, and genericity
AU - Garreta, Albert
AU - Miasnikov, Alexei
AU - Ovchinnikov, Denis
N1 - Publisher Copyright:
© 2020 Elsevier Inc.
PY - 2020/8/15
Y1 - 2020/8/15
N2 - We study finitely generated nilpotent groups G given by full rank finite presentations 〈A|R〉Nc in the variety Nc of nilpotent groups of class at most c, where c≥2. We prove that if the deficiency |A|−|R| is at least 2 then the group G is virtually free nilpotent, it is quasi finitely axiomatizable (in particular, first-order rigid), and it is almost (up to finite factors) directly indecomposable. One of the main results of the paper is that the Diophantine problem in nilpotent groups given by full rank finite presentations 〈A|R〉Nc is undecidable if |A|−|R|≥2 and decidable otherwise. We show that this class of groups is rather large since finite presentations asymptotically almost surely have full rank, so a random nilpotent group in the few relators model has a full rank presentation asymptotically almost surely. Full rank presentations give one a useful tool to approach random nilpotent groups and study their properties. Note, that the results above significantly improve our understanding of the Diophantine problem in finitely generated nilpotent groups: from a few special examples of groups with undecidable Diophantine problem we got to the place where we know that the Diophantine problem in all “typical” nilpotent groups is also undecidable.
AB - We study finitely generated nilpotent groups G given by full rank finite presentations 〈A|R〉Nc in the variety Nc of nilpotent groups of class at most c, where c≥2. We prove that if the deficiency |A|−|R| is at least 2 then the group G is virtually free nilpotent, it is quasi finitely axiomatizable (in particular, first-order rigid), and it is almost (up to finite factors) directly indecomposable. One of the main results of the paper is that the Diophantine problem in nilpotent groups given by full rank finite presentations 〈A|R〉Nc is undecidable if |A|−|R|≥2 and decidable otherwise. We show that this class of groups is rather large since finite presentations asymptotically almost surely have full rank, so a random nilpotent group in the few relators model has a full rank presentation asymptotically almost surely. Full rank presentations give one a useful tool to approach random nilpotent groups and study their properties. Note, that the results above significantly improve our understanding of the Diophantine problem in finitely generated nilpotent groups: from a few special examples of groups with undecidable Diophantine problem we got to the place where we know that the Diophantine problem in all “typical” nilpotent groups is also undecidable.
KW - Diophantine problem
KW - Free nilpotent group
KW - Nilpotent group
KW - Random group
KW - Random walk
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U2 - 10.1016/j.jalgebra.2020.01.030
DO - 10.1016/j.jalgebra.2020.01.030
M3 - Article
AN - SCOPUS:85082726136
SN - 0021-8693
VL - 556
SP - 1
EP - 34
JO - Journal of Algebra
JF - Journal of Algebra
ER -