Abstract The function F G ⁢ ( n ) {\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for F N ⁢ ( n ) {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and consequently, the statement of the asymptotic characterization of F N ⁢ ( n ) {\mathrm{F}_{N}(n)} is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for F N ⁢ ( n ) {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of the above article can be improved. Finally, we construct a class of finitely generated nilpotent groups N for which the asymptotic behavior of F N ⁢ ( n ) {\mathrm{F}_{N}(n)} can be fully characterized.

中文翻译：

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幂零群的残差维数

摘要 函数 FG ⁢ ( n ) {\mathrm{F}_{G}(n)} 给出了区分 G 的非平凡元素与具有满射群态射的恒等式所需的有限群的最大阶数。字长至多 n 的非平凡元素。在以前的工作中[M. Pengitore，有限生成幂零群的有效可分离性，纽约 J. 数学。24 2018, 83–145]，作者声称当 N 是有限生成的幂零群时 FN ⁢ ( n ) {\mathrm{F}_{N}(n)} 的表征。然而，上述主张的反例被传达给了作者，因此，FN ⁢ ( n ) {\mathrm{F}_{N}(n)} 的渐近表征的陈述是不正确的。在本文中，当 N 是有限生成的幂零群时，我们引入了新的工具来为 FN ⁢ ( n ) {\mathrm{F}_{N}(n)} 提供下渐近界。此外，我们引入了一类有限生成的幂零群，可以改进上述文章的上限。最后，我们构造了一类有限生成的幂零群 N，FN ⁢ ( n ) {\mathrm{F}_{N}(n)} 的渐近行为可以完全表征。