Residual Finiteness Growth in Virtually Abelian Groups
Jonas Deré, Joren Matthys
TL;DR
The paper exactly determines the asymptotic residual finiteness growth $ ext{RF}_G$ for finitely generated virtually abelian groups, showing it equals a power of a logarithm, $\text{RF}_G\asymp \log^k$, where $k$ is the maximal dimension of complex irreducible subrepresentations of the holonomy representation $\varphi: H\to \mathrm{GL}(m,\mathbb{Z})$ acting on a rank $m$ torsion-free abelian subgroup $K$ of finite index. The approach reduces the problem to the abelian lattice $K$ and its finite quotient action, then proves an upper bound via prime-density arguments and a lower bound via a sequence of reductions to $\mathbb{Q}$-irreducible components, Galois descent, and commuting-matrix analysis, culminating in $\text{RF}_G\approx \log^k$ with $0\le k\le m$. The results are shown to be sharp: for every $m\ge1$ and every $1\le k\le m$ there exist examples $G=\mathbb{Z}^m\rtimes H$ with $\text{RF}_G\approx \log^k$, and the paper also provides a practical method to compute $k$ from the character table of $H$. Applications include realizing all intermediate powers of $\log$ and illustrating non-quasi-isometric behavior among virtually abelian groups with the same abelian rank. Open questions address extending these exact asymptotics to broader virtually nilpotent classes and understanding the dependence on complex completions or Mal'cev structures.
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 $\text{RF}_G \approx \log^k$.