Residual Finiteness Growth in Virtually Nilpotent Groups

Abstract

The residual finiteness growth ${\text{RF}}_G:\mathbb{N}\to \mathbb{N}$ of a finitely generated group $G$ is a function that gives the smallest value of the index $[G:N]$ with $N$ a normal subgroup not containing a non-trivial element $g$, in function of the word norm of that element $g$. It has been studied for several classes of finitely generated groups, including free groups, linear groups and virtually abelian groups. This paper shows that if $G$ is virtually nilpotent, then ${\text{RF}}_G={\log }^{\hat{I}}´$ for some $\hat{I}´\in \mathbb{N}\cup \{0\}$, with moreover an explicit formula for $\hat{I}´$ in terms of Lie algebras. This implies in particular that it is an invariant of the complex Mal'cev completion, leading to the application that residual finiteness growth is a profinite invariant for virtually nilpotent groups.

25 pages