Aspherical Relative Presentations All Over Again

TL;DR

This paper reviews and consolidates key results on the concept of asphericity in relative group presentations, updating definitions, techniques, and classifications, especially focusing on one-relator cases with relator length four.

Contribution

It provides a comprehensive survey and unification of the theory of asphericity in relative presentations, including new applications and detailed classifications.

Findings

Updated definitions and terminology for asphericity.

Techniques for proving asphericity and non-asphericity.

Survey of results for relators with free product length four.

Abstract

The concept of asphericity for relative group presentations was introduced twenty five years ago. Since then, the subject has advanced and detailed asphericity classifications have been obtained for various families of one-relator relative presentations. Through this work the definition of asphericity has evolved and new applications have emerged. In this article we bring together key results on relative asphericity, update them, and exhibit them under a single set of definitions and terminology. We describe consequences of asphericity and present techniques for proving asphericity and for proving non-asphericity. We give a detailed survey of results concerning one-relator relative presentations where the relator has free product length four.