Survey on effective separability

TL;DR

This survey reviews the current state of effective separability in group theory, discussing how subsets are detected in finite quotients and highlighting open problems across various group classes.

Contribution

It provides a comprehensive overview of effective separability, combining classical and recent perspectives, and outlines open research questions in the field.

Findings

Summarizes known results on effective separability in different group classes.

Identifies key open problems and directions for future research.

Connects separability properties to algorithmic problems in groups.

Abstract

Separability for groups refers to the question which subsets of a group can be detected in its finite quotients. Classically, separability is studied in terms of which classes have a certain separability property, and this question is related to algorithmic problems in groups such as the word problem. A more recent perspective tries to study the order of the smallest finite quotient in which one detects the subset under consideration depending on its complexity, measured using the word norm on a finitely generated group. In this survey, we present what is currently known in the field of effective separability and give an overview of the open questions for several classes of groups.