A new test for asphericity and diagrammatic reducibility of group presentations

TL;DR

This paper introduces a new test for asphericity and diagrammatic reducibility of group presentations, extending classical methods and applying it to solve problems in group theory and related conjectures.

Contribution

It presents a novel test that surpasses classical weight tests, generalizes existing results, and addresses conjectures in group theory.

Findings

The new test proves diagrammatic reducibility where classical methods fail.

Generalizes results on asphericity of LOTs and Adian presentations.

Provides partial solutions to a conjecture of S.V. Ivanov.

Abstract

We present a new test for studying asphericity and diagrammatic reducibility of group presentations. Our test can be applied to prove diagrammatic reducibility in cases where the classical weight test fails. We use this criterion to generalize results of J. Howie and S.M. Gersten on asphericity of LOTs and of Adian presentations, and derive new results on solvability of equations over groups. We also use our methods to investigate a conjecture of S.V. Ivanov related to Kaplansky's problem on zero divisors: we strengthen Ivanov's result for locally indicable groups and prove a weak version of the conjecture.