Asphericity of length six relative group presentations

Abstract

Combinatorial group theory is a part of group theory that deals with groups given by presentations in terms of generators and defining relations. Many techniques both algebraic and geometric are used in dealing with problems in this area. In this thesis, we adopt the geometric approach. More specifically, we use so-called pictures over relative presentations to determine the asphericity of such presentations. We remark that if a relative presentation is aspherical then group theoretic information can be deduced. In Chapter 1, the concept of relative presentations is introduced and we state the main theorems and some known results. In Chapter 2, the concept of pictures is introduced and methods used for checking asphericity are explained. Excluding four unresolved cases, the asphericity of the relative presentation $\mathcal{P}$= $\langle G, x|x^{m}gxh\rangle$ for $m\geq2$ is determined in Chapter 3. If $H=\langle g, h\rangle$ $\leq G$, then the unresolved cases occur when $H$ is isomorphic to $C_{5}$ or $C_{6}$. The main work is done in Chapter 4, in which we investigate the asphericity of the relative presentation $\mathcal{P}$= $\langle G, x|xaxbxcxdxexf\rangle$, where the coefficients $a, b, c, d, e, f\in G$ and $x \notin G$ and prove the theorems stated in Chapter 1