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 ${\mathbb{H}}^2$. However, we give an example of substitutive tiling of $\Hyp^2$ which is non-primitive.

arXiv comment30 pages, 13 figures