Example: Why the fixed presentation matters

If a chart transition is $t={\mathcal{A}}_g{{\mathcal{A}}_f}^{-1}$, replacing ${\mathcal{A}}_g$ by $B_g{\mathcal{A}}_g$ gives $B_{g}t$. Closed words undergo the corresponding base-chart conjugation, and the geometric structure may be unchanged. Thus different valid chart presentations can have different transition maps. Peripheral holonomy also does not determine edge lengths or face shapes.