  A family of Poisson structures, parametrised by an arbitrary odd periodic
function $\phi$, is defined on the space $\cW$ of twisted polygons in
$\RR^\nu$. Poisson reductions with respect to two Poisson group actions on
$\cW$ are described. The $\nu=2$ and $\nu=3$ cases are discussed in detail and
the general $\nu$ case in less 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. A general result
is proved showing that, for any $\nu$, to certain concrete choices of $\phi$
there correspond compatible Poisson structures which generate all the extended
bigraded Toda hierarchies of a suitable size.
