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A Poisson structure is defined on the space   ${\mathcal {W}}$   of twisted polygons in   ${\mathbb {R}}^{\nu }$  . Poisson reductions with respect to two Poisson group actions on   ${\mathcal {W}}$   are described. The   $\nu =2$   and   $\nu =3$   cases are discussed in detail. Amongst the Poisson structures arising in examples are to be found the lattice Virasoro structure, the second Toda lattice structure, and some extended Toda lattice structures. It is shown that, in general, for any   $\nu$  , the reduction procedure gives rise to a family of bi-Hamiltonian structures on the reduced space.

Poisson Reduction of the Space of Polygons