Perturbation Effects on Word Lengths in Three-Reflection Symmetric Presentations of Dihedral Groups

We investigate the properties of word lengths of elements from a three-reflection symmetric generating set of the dihedral group $D_n$. Specifically, we provide the upper bound ${\lambda }_1(D_n,S)\le \lfloor \frac{n}{2}\rfloor +1$ for a quantity ${\lambda }_1$ defined in arXiv:1104.5044, which measures the stability of a finitely presented group under perturbations in the words corresponding to certain elements with respect to specific presentations. This quantity has been of recent interest due to its role in the application of group theory to computational genomics, and we aim to introduce techniques in additive combinatorics to its discourse.