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Tadpole Labelled Oriented Graph Groups and Cyclically Presented Groups

Howie, J and Williams, G (2012) 'Tadpole Labelled Oriented Graph Groups and Cyclically Presented Groups.' Journal of Algebra, 371. pp. 521-535. ISSN 0021-8693

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Abstract

We study a class of Labelled Oriented Graph (LOG) group where the underlying graph is a tadpole graph. We show that such a group is the natural HNN extension of a cyclically presented group and investigate the relationship between the LOG group and the cyclically presented group. We relate the second homotopy groups of their presentations and show that hyperbolicity of the cyclically presented group implies solvability of the conjugacy problem for the LOG group. In the case where the label on the tail of the LOG spells a positive word in the vertices in the circuit we show that the LOGs and groups coincide with those considered by Szczepa�nski and Vesnin. We obtain new presentations for these cyclically presented groups and show that the groups of Fibonacci type introduced by Johnson and Mawdesley are of this form. These groups generalize the Fibonacci groups and the Sieradski groups and have been studied by various authors. We continue these investigations, using small cancellation and curvature methods to obtain results on hyperbolicity, automaticity, SQ-universality, and solvability of decision problems.

Item Type: Article
Uncontrolled Keywords: Labelled Oriented Graph (LOG) group; HNN extension; Cyclically presented group; Fibonacci group; small cancellation theory; hyperbolic group; SQ-universal; decision problems
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science and Health
Faculty of Science and Health > Mathematical Sciences, Department of
SWORD Depositor: Elements
Depositing User: Elements
Date Deposited: 04 Sep 2012 08:31
Last Modified: 06 Jan 2022 13:27
URI: http://repository.essex.ac.uk/id/eprint/3799

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