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The symplectic geometry of polygons in hyperbolic 3-space

Michael Kapovich·John J. Millson·Thomas Treloar
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Abstract

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.

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Michael Kapovich, John J. Millson, Thomas Treloar. 2000-08-21. The symplectic geometry of polygons in hyperbolic 3-space. https://arxiv.org/abs/math/9907143

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