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Residual Finiteness Growth in Virtually Nilpotent Groups

Jonas Deré, Joren Matthys, Lukas Vandeputte

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.

Residual Finiteness Growth in Virtually Nilpotent Groups

Abstract

The residual finiteness growth of a finitely generated group is a function that gives the smallest value of the index with a normal subgroup not containing a non-trivial element , in function of the word norm of that element . 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 is virtually nilpotent, then for some , with moreover an explicit formula for 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.

Paper Structure

This paper contains 14 sections, 40 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie Groups and Algebras
  4. Residual Finiteness Growth
  5. Commensurable nilpotent groups
  6. Residual finiteness growth of nilpotent Lie rings
  7. From Ideal to Normal Subgroup
  8. From Normal Subgroup to Ideal
  9. Proof of Theorem \ref{['thm_intro_RFG_RFL']}
  10. Calculation of $\mathop{\mathrm{RF}}\nolimits_L$
  11. Definition of $\delta$
  12. The ideal $I_\delta$
  13. Proof of Theorem \ref{['thm_intro_exact']}
  14. Profinite invariance

Key Result

Theorem A

Let $\Gamma$ be a finitely generated virtually nilpotent group, then there exists $\delta \in \mathbb{N} \cup \{0\}$ such that $\mathop{\mathrm{RF}}\nolimits_\Gamma = \log^\delta$.

Theorems & Definitions (92)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 82 more