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

TL;DR

This paper studies how word lengths in dihedral groups are affected by perturbations, providing bounds and applying additive combinatorics to understand stability in group presentations relevant to computational genomics.

Contribution

It introduces an upper bound for a stability measure in dihedral groups with three-reflection symmetric generators, connecting group theory with additive combinatorics.

Findings

Established an upper bound for the stability measure $mbda_1(D_n,S)$

Applied additive combinatorics techniques to group presentation analysis

Relevance to computational genomics and group stability studies

Abstract

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 $\lambda_1(D_n,S) \leq \lfloor\frac{n}{2}\rfloor + 1$ 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.