The symplectic geometry of polygons in euclidean space

Abstract

We study the symplectic geometry of moduli spaces Mr of polygons with fixed side lengths in Euclidean space. We show that Mr has a natural structure of a complex analytic space and is complex-analytically isomorphic to the weighted quotient of (s2) n by PLS(2, C) constructed by Deligne and Mostow. We study the Hamiltonian flows on Mr obtained by bending the polygon along diagonals and show the group generated by such flows acts transitively on Mr. We also relate these flows to the twist flows of Goldman and Jeffrey-Weitsman. © 1996 J. differential geometry.