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Waring problems across algebra

Matej Brešar, Consuelo Mart\' inez

Abstract

The paper surveys various Waring type problems in groups, Lie algebras, and associative algebras.

Waring problems across algebra

Abstract

The paper surveys various Waring type problems in groups, Lie algebras, and associative algebras.
Paper Structure (14 sections, 26 theorems, 21 equations)

This paper contains 14 sections, 26 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Waring problems in groups
  3. Finite groups
  4. Verbal ellipticity of groups
  5. pro-$p$ groups
  6. Waring problems in Lie algebras
  7. Lie algebras linked to pro-$p$ groups
  8. Finitely generated nil algebras
  9. Waring problems in associative algebras
  10. Waring problems for general polynomials
  11. The L'vov-Kaplansky conjecture
  12. The commutator width
  13. The width of the product of commutators
  14. Multiplicative Waring problems

Key Result

Theorem 2.2

Let $\Gamma$ be a finitely generated residually-p torsion group and $G = \Gamma_p$ its pro-$p$-completion. Then an arbitrary word $w \in F$ is elliptic on $G$.

Theorems & Definitions (32)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Theorem 3.1
  • Theorem 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 22 more