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The Formanek-Procesi group with base a right-angled Artin group: Residual nilpotence and Lie algebra

V. Metaftsis, A. I. Papistas

Abstract

We investigate the Lie algebra of the Formanek-Procesi group FP(H) with base group H a right-angled Artin group. We show that the Lie algebra gr(FP(H)) has a presentation that is dictated by the group presentation. Moreover we show that FP(H) is a Magnus group.

The Formanek-Procesi group with base a right-angled Artin group: Residual nilpotence and Lie algebra

Abstract

We investigate the Lie algebra of the Formanek-Procesi group FP(H) with base group H a right-angled Artin group. We show that the Lie algebra gr(FP(H)) has a presentation that is dictated by the group presentation. Moreover we show that FP(H) is a Magnus group.

Paper Structure

This paper contains 11 sections, 32 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The elimination theorem
  4. Decomposition of ${\rm gr}(F_{n+1})$
  5. Analysis of $J$
  6. Free partially commutative Lie algebra
  7. Presentation of ${\rm gr}({\rm FP(H)})$
  8. Further analysis of $J$
  9. Analysis of $M$
  10. Main result
  11. Residual nilpotence of ${\rm FP(H)}$

Key Result

Theorem 1.1

${\rm gr}({\rm FP(H)}) \cong {\rm gr}(F_{n+1})/J$ as Lie algebras in a natural way. Moreover ${\rm FP(H)}$ is a Magnus group.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 2.1: Elimination
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Proposition 2.7
  • Proposition 3.1
  • Corollary 3.2
  • ...and 22 more