Residual Finiteness Growth in Virtually Abelian Groups

TL;DR

This paper investigates the residual finiteness growth in virtually abelian groups, establishing that it behaves like a power of logarithm, and constructs groups with specified growth rates.

Contribution

It provides the first explicit characterization of residual finiteness growth in virtually abelian groups and constructs examples with prescribed growth behaviors.

Findings

Residual finiteness growth in virtually abelian groups is approximately a logarithmic power.

Explicit formula for the growth rate exponent $k$ in terms of group parameters.

Existence of groups with normal abelian subgroups of rank $m$ and residual finiteness growth $oxed{ ext{log}^k}$ for $1 \\leq k \\leq m$.

Abstract

A group $G$ is called residually finite if for every non-trivial element $g \in G$, there exists a finite quotient $Q$ of $G$ such that the element $g$ is non-trivial in the quotient as well. Instead of just investigating whether a group satisfies this property, a new perspective is to quantify residual finiteness by studying the minimal size of the finite quotient $Q$ depending on the complexity of the element $g$, for example by using the word norm $\|g\|_G$ if the group $G$ is assumed to be finitely generated. The residual finiteness growth $\text{RF}_G: \mathbb{N} \to \mathbb{N}$ is then defined as the smallest function such that if $\|g\|_G \leq r$, there exists a morphism $\varphi: G \to Q$ to a finite group $Q$ with $|Q| \leq \text{RF}_G(r)$ and $\varphi(g) \neq e_Q$. Although upper bounds have been established for several classes of groups, exact asymptotics for the function $\text{RF}_G$ are only known for very few groups such as abelian groups, the Grigorchuk group and certain arithmetic groups. In this paper, we show that the residual finiteness growth of virtually abelian groups equals $\log^k$ for some $k \in \mathbb{N}$, where the value $k$ is given by an explicit expression. As an application, we show that for every $m \geq 1$ and every $1 \leq k \leq m$, there exists a group $G$ containing a normal abelian subgroup of rank $m$ and with $\text{RF}_G \approx \log^k$.