The Stable Limit of Moduli Spaces of Polygons

Polygon spaces have been studied extensively, and yet missing from the literature is a simple property that every polygon has: dimension. This is distinct (possibly) from the dimension of the ambient space in which the polygon lives. A square, in the usual sense of the word, is $2$-dimensional no matter the dimension of the ambient space in which it is embedded. If the ambient space has dimension greater than or equal to $3$ we may bend the square along a diagonal to produce a $3$-dimensional polygon with the same edge-lengths. And yet even if the dimension of the ambient space is large, no amount of bending of the square will produce a polygon of dimension larger than $3$. We generalize this idea to show that there are only finitely many moduli spaces of polygons with given edge-lengths, even as the ambient dimension increases without bound.