On the asphericity of length five relative group presentations (Q2766389)

A relative group presentation, closely related to the notion of equations over a group \(G\), is an expression of the form \(\langle G,{\mathbf x}\mid{\mathbf r}\rangle\) where \(G\) is a group, \(\mathbf x\) a set disjoint from \(G\), and \(\mathbf r\) a set of cyclically reduced words in the free product \(G*\langle\mathbf x\rangle\) of \(G\) and the free group on \(\mathbf x\). The group defined by the presentation is \((G*\langle\mathbf x\rangle)/N\) where \(N\) is the normal closure of \(\mathbf r\). The presentation is injective if \(G\) injects into this group, and aspherical if every spherical picture over it contains a dipole (equivalently, the second homotopy group of the relative 2-complex \((Y,X)\) associated to the presentation, where \(X\) is a \(K(G,1)\)-space, is generated by some simple type of elements). Every aspherical relative group presentation is injective, but the converse is far from true. F. Levin showed that any relative group presentation of the form \(\langle G,t\mid g_1tg_2t\cdots g_kt=1\rangle\) is injective. The classification of the aspherical ones among these presentations is known for \(k=2\), 3 and 4, and the main result of the present paper concerns the case \(k=5\): it is shown that, for \(k=5\), the presentation is aspherical if the subgroup of \(G\) generated by \(g_ig_{i+1}^{-1}\), \(1\leq i\leq 4\), is neither finite cyclic nor a finite triangle group. The paper finishes with a discussion of the open cases which the authors are unable to treat.