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A new test for asphericity and diagrammatic reducibility of group presentations

Part of: Special aspects of infinite or finite groups Low-dimensional topology

Published online by Cambridge University Press:  26 January 2019

Jonathan Ariel Barmak  and
Elias Gabriel Minian
Jonathan Ariel Barmak
Affiliation:
Departamento de Matemática–IMAS FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina (jbarmak@dm.uba.ar; gminian@dm.uba.ar)
Elias Gabriel Minian
Affiliation:
Departamento de Matemática–IMAS FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina (jbarmak@dm.uba.ar; gminian@dm.uba.ar)
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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.

Keywords

AsphericityDR presentationslabelled oriented treesweight testlocally indicable groupsAdian presentations

MSC classification

Primary: 57M20: Two-dimensional complexes
Secondary: 20F05: Generators, relations, and presentations 20F06: Cancellation theory; application of van Kampen diagrams 57M05: Fundamental group, presentations, free differential calculus
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 871 - 895
Copyright
Copyright © Royal Society of Edinburgh 2019

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