So, if I start with a full Dihedral group D2n to represent a regular, ideal polygon in the hyperbolic plane, then I remove an element (and any subsequently necessary elements so that it is still a group) so that now it still represents a polygon, but that is not regular, is it possible to generalize the geometric effects on the polygon?

Like, if restricting to specific subgroups of D2n, or removal of specific elements will always produce the same geometric effect in the polygon (like an increase in surface area, or something).