Equationally separable classes of groups

TL;DR

This work develops a general method to separate natural classes of groups by finite systems of equations, introducing the notion of equational separability within a larger class. A key meta-theorem on simple classes, combined with a reduction of unknowns to a univariate system via relative-hyperbolicity constructions, yields separability results for several important classes (e.g., finite vs periodic, decidable conjugacy vs word problems, locally indicable vs left-orderable), with some cases permitting univariate separating systems. A notable application proves that countable amalgams of periodic groups with finite intersection embed into a periodic group, addressing a 1960 Neumann question in the countable setting. The results illuminate when equations over groups suffice to distinguish between classes and provide a framework (RH-closedness and quasi-identities) to extend univariate and finite-system separations to broader contexts, potentially impacting decision problems and group-embedding theory.

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.

Theorems & Definitions (31)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Proposition 2.2
• proof
• Theorem 2.3
• proof
• Lemma 2.4
• proof