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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).

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As far as I can see, "remove an element (and any subsequently necessary elements so that it is still a group)" is not well-defined, as there may be various choices for what to remove subsequently. The closest well-defined approximation that I can think of is "pass to a subgroup". Is that what you meant? Also, what do you mean by a subgroup "representing" a non-regular polygon? I'll vote to close as "not a real question" pending clarification. – Andreas Blass Jun 11 '13 at 19:22
Sorry for the lack of clarification. What I meant was, say you start with an ideal, regular square in the hyperbolic plane. Then i's symmetric group is all of D8. But what if we had another ideal quadrilateral whose symmetries were described by some subgroup of D8. based on the elements in (or rather, not in) this subgroup, is it possible to make a generalization as to how this polygon will look geometrically? Will it always be the case that the perimeter or surface area increase or decrease, or could we tell if two opposite sides had the same length,but not the other two, something like that. – Taylor Matyasz Jun 11 '13 at 19:48

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