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Equations in wreath products

Laurent Bartholdi, Ruiwen Dong, Leon Pernak, Jan Philipp Wächter

TL;DR

The paper investigates solvability of equations in wreath products, focusing on quadratic diophantine problems. It develops group-ring reductions and genus-based decompositions to establish decidability of the orientable quadratic diophantine problem in wreath products of finitely generated Abelian groups and to determine commutator width, proving $\lceil\operatorname{rank}(B)/2\rceil$ for $A\wr B$. It further shows solvability of $\mathcal{Q^+DP}$ in Baumslag's finitely presented metabelian group, highlighting the boundaries of decidability in metabelian contexts. The results extend the understanding of Diophantine problems in large, structured groups and connect algebraic properties to combinatorial and geometric interpretations via group rings and lattice methods.

Abstract

We survey solvability of equations in wreath products of groups, and prove that the quadratic diophantine problem is solvable in wreath products of Abelian groups. We consider the related question of determining commutator width, and prove that the quadratic diophantine problem is also solvable in Baumslag's finitely presented metabelian group. This text is a short version of an extensive article by the first-named authors.

Equations in wreath products

TL;DR

The paper investigates solvability of equations in wreath products, focusing on quadratic diophantine problems. It develops group-ring reductions and genus-based decompositions to establish decidability of the orientable quadratic diophantine problem in wreath products of finitely generated Abelian groups and to determine commutator width, proving for . It further shows solvability of in Baumslag's finitely presented metabelian group, highlighting the boundaries of decidability in metabelian contexts. The results extend the understanding of Diophantine problems in large, structured groups and connect algebraic properties to combinatorial and geometric interpretations via group rings and lattice methods.

Abstract

We survey solvability of equations in wreath products of groups, and prove that the quadratic diophantine problem is solvable in wreath products of Abelian groups. We consider the related question of determining commutator width, and prove that the quadratic diophantine problem is also solvable in Baumslag's finitely presented metabelian group. This text is a short version of an extensive article by the first-named authors.
Paper Structure (6 sections, 3 theorems, 8 equations)

This paper contains 6 sections, 3 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proofs
  4. Spherical equations
  5. Commutators in wreath products
  6. The Baumslag group

Key Result

Theorem 1

Let $A,B$ be two finitely generated Abelian groups. Then the quadratic diophantine problem ${\mathcal{Q^+DP}}$ is solvable in $A\wr B$.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3