  Given an n-dimensional substitution whose associated linear expansion is
unimodular and hyperbolic, we use elements of the one-dimensional integer
\v{C}ech cohomology of the associated tiling space to construct a finite-to-one
semi-conjugacy, called geometric realization, between the substitution induced
dynamics and an invariant set of a hyperbolic toral automorphism. If the linear
expansion satisfies a Pisot family condition and the rank of the module of
generalized return vectors equals the generalized degree of the linear
expansion, the image of geometric realization is the entire torus and coincides
with the map onto the maximal equicontinuous factor of the translation action
on the tiling space. We are led to formulate a higher-dimensional
generalization of the Pisot Substitution Conjecture: If the linear expansion
satisfies the Pisot family condition and the rank of the one-dimensional
cohomology of the tiling space equals the generalized degree of the linear
expansion, then the translation action on the tiling space has pure discrete
spectrum.
