LOG groups and cyclically presented groups
A labelled oriented graph (LOG) is a directed graph together with a labelling of its edges by vertices. It determines a group presentation with generators corresponding to the vertices, and relators determined by the edges where the relator says that the edge label conjugates the (generator corresponding to the) initial vertex of the edge to the terminal vertex. The Wirtinger presentation of a classical knot group (with the redundant relator removed) is an LOG presentation for which the underlying graph is a chain. More generally the fundamental group of the complement in \(S^{n+2}\) of any smoothly imbedded, closed, orientable \(n\)-manifold is an LOG group. In this paper, the authors study a class of LOG presentations for which the underlying graph consists of a circuit together with one other edge which has one endpoint a vertex in the circuit. These are endowed with a certain kind of labelling, and the resulting groups are denoted \(G(n,t)\) where \(n\) is the number of vertices in the circuit and \(t\) is determined by the particular labelling. Each \(G(n,t)\) is a semidirect product of its commutator subgroup \(H(n,t)\) with an infinite cyclic group. The \(H(n,t)\) have cyclic presentations analogous to those of the Fibonacci groups. For all but five values of the pair of parameters \((n,t)\), it is determined whether \(H(n,t)\) is finite, and for the finite cases the group is determined. The authors also study the asphericity of the 2-complex \(L(n,t)\) determined by the cyclic presentation of \(H(n,t)\). For most values of \((n,t)\) (including all values with \(n\geq 10\)) except for the families \(L(n,0)\), \(L(n,2)\), \(L(n,n-1)\), \(L(2k,k+1)\), and \(L(2t-1,t)\), \(L(n,t)\) is aspherical, and hence \(H(n,t)\) and \(G(n,t)\) are torsionfree.
scientific article; zbMATH DE number 783856
scientific article