Residual dimension of nilpotent groups

TL;DR

This paper investigates the residual finiteness growth of finitely generated nilpotent groups, providing new bounds and characterizations for the asymptotic behavior of the residual finiteness function.

Contribution

The paper introduces new tools for lower bounds, improves upper bounds for certain groups, and fully characterizes the asymptotic behavior of residual finiteness for a class of nilpotent groups.

Findings

Established new lower asymptotic bounds for residual finiteness functions.

Improved upper bounds for specific classes of finitely generated nilpotent groups.

Fully characterized the asymptotic behavior of residual finiteness for a particular class of nilpotent groups.

Abstract

The functions $F_{G}(n)$ measures the asymptotic behavior of residual finiteness for a finitely generated group $G$. In previous work \cite{Pengitore_1}, the author claimed a characterization for $F_{N}(n)$ when $N$ is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and subsequently, the statement of the asymptotic characterization of $F_{N}(n)$ is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for $F_{N}(n)$ when $N$ is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of \cite{Pengitore_1} can be improved. Finally, we construct of a class of finitely generated nilpotent groups $N$ for which the asymptotic behavior of $F_N(n)$ can be fully characterized.