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Generalized small cancellation conditions, non-positive curvature and diagrammatic reducibility

Part of: Special aspects of infinite or finite groups Semigroups Algebraic structures

Published online by Cambridge University Press:  02 March 2021

Martín Axel Blufstein ,
Elías Gabriel Minian  and
Iván Sadofschi Costa
Martín Axel Blufstein
Affiliation:
Departamento de Matemática - IMAS FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina (mblufstein@dm.uba.ar; gminian@dm.uba.ar; isadofschi@dm.uba.ar)
Elías Gabriel Minian
Affiliation:
Departamento de Matemática - IMAS FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina (mblufstein@dm.uba.ar; gminian@dm.uba.ar; isadofschi@dm.uba.ar)
Iván Sadofschi Costa
Affiliation:
Departamento de Matemática - IMAS FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina (mblufstein@dm.uba.ar; gminian@dm.uba.ar; isadofschi@dm.uba.ar)
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Abstract

We present a metric condition $\TTMetric$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition $\TTMetric$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular, we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, $\TTMetric$-groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation P = 〈X| R〉 of group G satisfies conditions $\TTMetric -C'(\frac {1}{2})$, the length of any nontrivial word in the free group generated by X representing the trivial element in G is at least that of the shortest relator. We also introduce a strict metric condition $\TTMetricStrict$, which implies hyperbolicity.

Keywords

Small cancellation theoryword problemDehn functionArtin groupsnon-positive curvatureconjugacy problemhyperbolic groupsisoperimetric inequality

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 20F67: Hyperbolic groups and nonpositively curved groups 20F06: Cancellation theory; application of van Kampen diagrams 20F10: Word problems, other decision problems, connections with logic and automata 08A50: Word problems 20M05: Free semigroups, generators and relations, word problems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 3 , June 2022 , pp. 545 - 566
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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