Word Length Perturbations in Certain Symmetric Presentations of Dihedral Groups

Given a finite group, $G$, with a generating subset $S$, there is a well-established notion of length for an element $g \in G$ given in terms of the minimal length expression for $g$ as a product of elements from $S$. Recently, quantities denoted $\lambda_{1}(G,S)$ and $\lambda_{2}(G,S)$ have been defined that allow for a precise measure of how stable a group is under certain types of small perturbations in the generating expressions for the elements of the group. In this paper, we further expose the fundamental properties of $\lambda_{1}(G,S)$ and $\lambda_{2}(G,S)$ by establishing their bounds when $G=D_{n}$, a dihedral group. An essential step in our approach is to completely characterize so-called symmetric presentations of the groups $D_{n}$, providing insight into the manner in which $\lambda_{1}(G,S)$ and $\lambda_{2}(G,S)$ interact with finite group presentations. This is of interest independent of the study of the quantities $\lambda_{1}(G,S),\; \lambda_{2}(G,S)$. Finally, we discuss several conjectures and open questions for future consideration.