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Amalgams of rational unipotent groups and residual nilpotence

Pierre-Emmanuel Caprace, Timothée Marquis

Abstract

We provide sufficient conditions for a free amalgamated product of torsionfree nilpotent groups to be residually nilpotent. We also characterise the residual nilpotence of certain higher-dimensional amalgams of unipotent groups over the rationals (known as KMS groups) in terms of their defining Cartan matrix. As an application, we give a normal form for the elements of a minimal Kac-Moody group over the rationals.

Amalgams of rational unipotent groups and residual nilpotence

Abstract

We provide sufficient conditions for a free amalgamated product of torsionfree nilpotent groups to be residually nilpotent. We also characterise the residual nilpotence of certain higher-dimensional amalgams of unipotent groups over the rationals (known as KMS groups) in terms of their defining Cartan matrix. As an application, we give a normal form for the elements of a minimal Kac-Moody group over the rationals.
Paper Structure (20 sections, 31 theorems, 80 equations)

This paper contains 20 sections, 31 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Residual nilpotence of free amalgamated products
  3. A triangle of groups with rational Heisenberg vertex groups
  4. Amalgams of rational unipotent groups
  5. Preliminaries
  6. Group commutators
  7. Kac--Moody algebras
  8. Minimal Kac--Moody groups
  9. Maximal Kac--Moody groups
  10. Kac--Moody--Steinberg groups
  11. Coset graphs
  12. Spherical rank $2$ subgroups
  13. Free amalgamated products
  14. Groups generated by copies of $\mathbb{Q}$
  15. Pronilpotent completions
  16. ...and 5 more sections

Key Result

Theorem 1

The free amalgamated product $A*_C B$ is residually nilpotent as soon as the following conditions hold:

Theorems & Definitions (59)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Corollary 6
  • Corollary 7
  • Remark 2.1
  • Lemma 3.1
  • proof
  • ...and 49 more