Residual Finiteness Growth in Two-Step Nilpotent Groups

TL;DR

The paper investigates the residual finiteness growth $\mathrm{RF}_G$ for finitely generated residually finite groups, with a focus on finitely generated two-step nilpotent groups. By expressing these groups through an alternating bilinear map $\varphi$ (via $G_\varphi$) and relating $\mathrm{RF}_G$ to the complex Mal'cev completion $G^{\mathbb{C}}$, the authors prove an improved upper bound $\mathrm{RF}_{G_\varphi} \preceq \log^{\,d(\varphi^{\mathbb{C}})+1}$ and establish exactness in low-rank cases, leading to a conjecture that this bound is always sharp. They develop a lower bound tied to the ideals $I_d(M_x)$ and show that, in many instances, the upper and lower bounds coincide, notably for $\mathcal{I}(m,1)$ and $\mathcal{I}(m,2)$ groups. The results support the conjecture that $\mathrm{RF}_G$ can be controlled by the complex Mal'cev completion and provide insights into when $\mathrm{RF}_G$ is a quasi-isometric invariant within this class. Overall, the work connects residual finiteness growth to Lie-algebraic properties and primes reductions, yielding precise polylogarithmic growth for significant families of 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 $φ: G \to Q$ with $φ(g) \neq e_Q$ 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.

Figures (1)

• Figure 1: Example illustrating the construction of the tuples of Lemma \ref{['lem_distinct_tuples']}. There is one subset of one element and two subsets with two elements, hence the elements can be represented as the set $\{(1,1); (2,1); (2,2); (3,1); (3,2)\}$, yielding the tree above. Now, the first tuple of this tree is $\{(1,1); (2,1); (2,2)\}$. It has the first two entries in common with the second tuple $\{(1,1); (2,1); (3,1)\}$, but the last entry comes from a different subset.

Theorems & Definitions (116)

• Conjecture 1
• Theorem 1.1
• Proposition 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• Definition 2.6
• Definition 2.7