Residual Finiteness Growth in Two-Step Nilpotent Groups

Abstract

Given a finitely generated residually finite group $G$, the residual finiteness growth $\text{RF}_G: \mathbb{N} \to \mathbb{N}$ bounds the size of a finite group $Q$ needed to detect an element of norm at most $r$. More specifically, if $g\in G$ is a non-trivial element with $\|g\|_G \leq r$, so $g$ can be written as a product of at most $r$ generators or their inverses, then we can find a homomorphism $\phi: G \to Q$ with $\phi(g) \neq e_Q$ and $|Q| \leq \text{RF}_G(r)$. The residual finiteness growth is defined as the smallest function with this property. This function has been bounded from above and below for several classes of groups, including virtually abelian, nilpotent, linear and free groups. However, for many of these groups, the exact asymptotics of $\text{RF}_G$ are unknown (in particular this is the case for a general nilpotent group), nor whether it is a quasi-isometric invariant for certain classes of groups. In this paper, we make a first step in giving an affirmative answer to the latter question for $2$-step nilpotent groups, by improving the polylogarithmic upper bound known in literature, and to show that it only depends on the complex Mal'cev completion of the group. If the commutator subgroup is one- or two-dimensional, we prove that our bound is in fact exact, and we conjecture that this holds in general.