Equationally separable classes of groups

TL;DR

This paper investigates classes of groups defined by equations, showing how certain properties allow solutions in larger groups and embedding results, thus advancing understanding of group equations and structures.

Contribution

It introduces new results on equationally separable classes of groups and solves a longstanding problem about embedding amalgams of periodic groups.

Findings

Existence of finite systems of equations with specific solution properties in various group classes

Embedding of amalgams of countable periodic groups into larger periodic groups

Answers a 1960 question of B. Neumann regarding group embeddings

Abstract

Over each nontrivial finite group $G$, there exists a finite system of equations having no solutions in larger finite groups but having a solution in a periodic group containing $G$. We prove several similar facts about amenable, orderable, locally indicable, solvable, nilpotent, and other classes of groups. As a byproduct, we also show that any amalgam of two countable periodic groups with finite intersection embeds into a periodic group, thereby answering a 1960 question of B. Neumann in the countable case.