Residual Finiteness Growth in Virtually Nilpotent Groups

TL;DR

This paper characterizes the residual finiteness growth of finitely generated virtually nilpotent groups, showing it follows a logarithmic power law and is a profinite invariant, with explicit formulas derived from Lie algebra structures.

Contribution

It provides an explicit formula for the residual finiteness growth in virtually nilpotent groups and establishes it as a profinite invariant.

Findings

Residual finiteness growth is logarithmic power law in virtually nilpotent groups.

Explicit formula for the growth rate based on Lie algebra data.

Residual finiteness growth is a profinite invariant for these 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 $\text{RF}_G = \log^\delta$ for some $\delta\in \mathbb{N}\cup\{0\}$, with moreover an explicit formula for $\delta$ 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.