Residual finiteness growth in two-step nilpotent groups

Abstract Given a finitely generated residually finite group 𝐺, the residual finiteness growth <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>RF</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:mi mathvariant="double-struck">N</m:mi> <m:mo stretchy="false">→</m:mo> <m:mi mathvariant="double-struck">N</m:mi> </m:mrow> </m:mrow> </m:math> \operatorname{RF}_{G}\colon\mathbb{N}\to\mathbb{N} bounds the size of a finite group 𝑄 needed to detect an element of norm at most 𝑟. This function has been estimated for several classes of groups, including virtually abelian, nilpotent, linear and free groups. However, for many of these groups, the exact asymptotics of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>RF</m:mi> <m:mi>G</m:mi> </m:msub> </m:math> \operatorname{RF}_{G} are unknown; in particular, this is the case for a general nilpotent group. In this paper, we improve the polylogarithmic upper bound known in the literature for 2-step nilpotent groups, and we 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.