AbstractThe function {\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 {\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 {\mathrm{F}_{N}(n)} is incorrect.
In this article, we introduce new tools to provide lower asymptotic bounds for {\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 {\mathrm{F}_{N}(n)} can be fully characterized.
Residual dimension of nilpotent groups
