Tadpole Labelled Oriented Graph groups and cyclically presented groups

JOURNAL OF ALGEBRA

卷:371

Article

Howie, James^2,3  Williams, Gerald¹ 

[1] Univ Essex, Dept Math Sci, Colchester CO4 3SQ, Essex, England

[2] Heriot Watt Univ, Dept Math, Edinburgh EH14 4AS, Midlothian, Scotland

[3] Heriot Watt Univ, Maxwell Inst Math Sci, Edinburgh EH14 4AS, Midlothian, Scotland

关键词: Labelled Oriented Graph (LOG) group;    HNN extension;    Cyclically presented group;    Fibonacci group;    Small cancellation theory;    Hyperbolic group;    SQ-universal;    Decision problems;   

DOI  :  10.1016/j.jalgebra.2012.09.001

来源: Elsevier

【 摘 要 】

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 Szczepanski 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 generalise 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. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free