The Stable Limit of Moduli Spaces of Polygons

TL;DR

This paper investigates the dimension of polygon moduli spaces, showing that for fixed edge lengths, only finitely many such spaces exist regardless of the ambient space's dimension, revealing a stable property.

Contribution

It introduces a new understanding of the dimension stability of polygon moduli spaces across varying ambient space dimensions.

Findings

Finitely many moduli spaces for fixed edge lengths exist.

Polygon dimension remains bounded regardless of ambient space dimension.

The result generalizes the concept of polygon flexibility in high-dimensional spaces.

Abstract

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.