Geometric realizations of two dimensional substitutive tilings

Abstract: We define 2-dimensional topological substitutions. A tiling of the Euclidean plane, or of the hyperbolic plane, is substitutive if the underlying 2-complex can be obtained by iteration of a 2-dimensional topological substitution. We prove that there is no primitive substitutive tiling of the hyperbolic plane $\mathbb{H}<sup>2$. However, we give an example of substitutive tiling of $\Hyp<sup>2$ which is non-primitive.