Residual Finiteness Growth in Minimax Groups
Jonas Deré, Joren Matthys
TL;DR
We address residual finiteness growth for finitely generated residually finite minimax groups, showing that for $G$ sitting in $1\to K \to G \to H \to 1$ with $K$ torsion-free nilpotent of Prüfer rank $m$ and $H$ virtually abelian of rank $n$, the growth satisfies $RF_G \preceq r^{m+4n}$. The approach blends Mal'cev theory, Lie-algebra methods, and Chebotarev-density arguments to construct finite quotients that separate elements within radius $r$ while keeping the quotient sizes controlled. The work delivers a sharp, extension-invariant upper bound and a linear lower bound for non-virtually-nilpotent groups, with polylog refinements for virtually nilpotent cases and sharp behavior in several subcases (virtually abelian, BS(1,$n$)). Overall, the results tighten the connection between the algebraic structure of minimax groups and their residual finiteness growth, improving on previous linear-in-dimension bounds for linear groups and clarifying when polylog or linear lower bounds appear.
Abstract
If $g\in G$ is a non-trivial element in a residually finite group, then there exists by definition a finite group $Q$ and a homomorphism $\varphi: G \to Q$ such that $\varphi(g) \neq e$. The residual finiteness growth $\text{RF}_G$ of a finitely generated residually finite group $G$ estimates the size of $Q$ in terms of the word norm $\|g\|$ of the element $g\in G$. This function has been studied for several classes of groups, including free groups, lamplighter groups and nilpotent groups. For finitely generated linear groups $G\leq \text{GL}(m, \mathbb{C})$ this function is known to be bounded by $\text{RF}_G(r) \preceq r^{m^2+1}$, which is quadratic in $m$. This paper establishes an improved bound of the form $\text{RF}_G(r) \preceq r^{4k}$ with $k$ the Prüfer rank of $G$ for certain virtually solvable linear groups, namely minimax groups, a class which includes virtually polycyclic and Baumslag-Solitar groups. Moreover, the upper bound is invariant under taking finite extensions, and also establishes an improved polylogarithmic version for virtually nilpotent groups, generalizing the known exact bound for virtually abelian groups. If the group is not virtually nilpotent, we prove that $\text{RF}_G(r)$ is at least linear, improving a recent result.