Abstract We study finitely generated nilpotent groups G given by full rank finite presentations 〈 A | R 〉 N c in the variety N c 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 〉 N c 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.

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满秩表示和幂零群：结构、丢番图问题和通用性

摘要 我们研究了由满秩有限表示给出的有限生成幂零群 G < A | R ＞ N c 在类至多 c 的幂零群的变体 N c 中，其中 c ≥ 2 。我们证明如果不足| 一个 | - | R | 至少为 2 则群 G 几乎是自由幂零的，它是准有限公理化的（特别是一阶刚性的），并且它几乎（直到有限因子）直接不可分解。该论文的主要结果之一是，幂零群中的丢番图问题由全秩有限表示 〈 A | R 〉 N c 是不可判定的，如果 | 一个 | - | R | ≥ 2，否则可确定。我们证明这类群相当大，因为渐近的有限表示几乎肯定有满秩，因此，少数关系模型中的随机幂零群几乎可以肯定地渐近地具有满秩表示。全秩演示为人们提供了一种有用的工具来处理随机幂零群并研究它们的特性。请注意，上述结果显着提高了我们对有限生成幂零群中丢番图问题的理解：从具有不可判定丢番图问题的群的几个特殊例子，我们到达了我们知道所有“典型”幂零群中的丢番图问题的地方也是不可判定的。