Residual Finiteness Growth in Minimax Groups

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 $φ: G \to Q$ such that $φ(g) \neq e$. 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.