Year: 2025
Type: preprint
Abstract: The residual finiteness growth $\text{RF}_G: \mathbb{N} \to \mathbb{N}$ of a finitely generated group $G$ is a function that gives the smallest value of the index $[G:N]$ with $N$ a normal subgroup not containing a non-trivial element $g$, in function of the word norm of that element $g$. It has been studied for several classes of finitely generated groups, including free groups, linear groups and virtually abelian groups. This paper shows that if $G$ is virtually nilpotent, then $\text{RF}_G = \log^δ$ for some $δ\in \mathbb{N}\cup\{0\}$, with moreover an explicit formula for $δ$ in terms of Lie algebras. This implies in particular that it is an invariant of the complex Mal'cev completion, leading to the application that residual finiteness growth is a profinite invariant for virtually nilpotent groups.
Source: arXiv (Cornell University)
Authors: Deré, Jonas, Matthys, Joren, Vandeputte, Lukas
Institution: KU Leuven
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: KU Leuven
Award: G436553921
OpenAlex ID: W4417530532
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