Asphericity of a length four relative group presentation

TL;DR

This paper characterizes the asphericity of a specific length four relative group presentation, identifying exact conditions and exceptional cases based on subgroup isomorphisms.

Contribution

It provides a complete classification of when the given relative presentation is aspherical, including precise exceptional cases based on subgroup structures.

Findings

Asphericity determined by subgroup isomorphism to certain cyclic groups.

Exact conditions for asphericity with a small set of exceptions.

Identification of exceptional cases when subgroup H is isomorphic to C_5, C_6, C_8, or C_2×C_4.

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 $C_2 \times C_4$.