Full residual finiteness growths of nilpotent groups

TL;DR

This paper investigates the full residual finiteness growth of finitely generated nilpotent groups, establishing precise growth rates under certain conditions and providing bounds and characterizations for the general case.

Contribution

It introduces a method to determine upper bounds for residual finiteness growth in nilpotent groups using terraced filtrations and characterizes when this growth matches word growth.

Findings

Growth is exactly $n^b$ when the last term of the lower central series has finite index in the center.

Provides a method to find an upper bound of the form $n^b$ for general nilpotent groups.

Characterizes nilpotent groups where residual finiteness growth equals word growth.

Abstract

Full residual finiteness growth of a finitely generated group $G$ measures how efficiently word metric $n$-balls of $G$ inject into finite quotients of $G$. We initiate a study of this growth over the class of nilpotent groups. When the last term of the lower central series of $G$ has finite index in the center of $G$ we show that the growth is precisely $n^b$, where $b$ is the product of the nilpotency class and dimension of $G$. In the general case, we give a method for finding an upper bound of the form $n^b$ where $b$ is a natural number determined by what we call a terraced filtration of $G$. Finally, we characterize nilpotent groups for which the word growth and full residual finiteness growth coincide.