  We study the symplectic geometry of the moduli space of closed n-gons with
fixed side-lengths in hyperbolic 3-space. We prove that these moduli spaces
have a symplectic structure coming from Poisson Lie theory. We construct
completely integrable systems on these moduli spaces by bending n-gons along
their diagonals. The results of this paper are the analogues of the results of
the first two authors for n-gon linkages in Euclidean 3-space in JDG 44. We
conclude by proving by a deformation argument that the moduli spaces of n-gon
linkages in hyperbolic 3-space and Euclidean 3-space with the same set of
side-lengths are symplectomorphic.
