Asphericity of a length four relative group presentation

Abstract

We consider the relative group presentation $\mathcal{P} = \langle G, \mathbf{x} | \mathbf{r} \rangle$ where $\mathbf{x} = \{ x \}$ and $\mathbf{r} = \{ xg_1 xg_2 xg_3 x^{-1} g_4 \}$. We show modulo a small number of exceptional cases exactly when $\mathcal{P}$ is aspherical. If $H = \langle g_1^{-1} g_2, g_1^{-1} g_3 g_1 , g_4 \rangle \leq G$ then the exceptional cases occur when $H$ is isomorphic to one of $C_5,C_6,C_8$ or $C_2 \times C_4$.