Asphericity of a length four relative group presentation

Abstract

We consider the relative group presentation $\mathcal{P}=⟨G,\mathbf{x}|\mathbf{r}⟩$ where $\mathbf{x}=\{x\}$ and $\mathbf{r}=\{x{g}_{1}x{g}_{2}x{g}_3{x}^{-1}{g}_4\}$. We show modulo a small number of exceptional cases exactly when $\mathcal{P}$ is aspherical. If $H=⟨g_1^{-1}{g}_2,g_1^{-1}{g}_3{g}_1,g_4⟩\le G$ then the exceptional cases occur when $H$ is isomorphic to one of $C_5,C_6,C_8$ or $C_2\times C_4$.

arXiv comment53 pages