Complexity and cohomology for cut-and-projection tilings

We consider a subclass of tilings: the tilings obtained by cut-and-projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponent a in terms of the ranks of certain groups which appear in the construction. We give bounds for a. These computations apply to some well-known tilings, such as the octagonal tilings, or tilings associated with billiard sequences. A link is made between the exponent of the complexity, and the fact that the cohomoloay of the associated tiling space is finitely generated over Q. We show that such a link cannot be established for more general filings, and we present a counterexample in dimension one.