Tadpole Labelled Oriented Graph Groups and Cyclically Presented Groups

TL;DR

This paper explores tadpole labelled oriented graph groups, revealing their connection to cyclically presented groups, and investigates properties like hyperbolicity, automaticity, and decision problem solvability using geometric and algebraic methods.

Contribution

It establishes that tadpole LOG groups are HNN extensions of cyclically presented groups and provides new presentations and properties for these groups, including those of Fibonacci type.

Findings

Hyperbolicity of cyclically presented groups implies conjugacy problem solvability in LOG groups.

New presentations for Fibonacci-type groups are derived.

Results on hyperbolicity, automaticity, and SQ-universality are obtained using small cancellation and curvature methods.

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 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 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.