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We consider a subclass of tilings, the tilings obtained by cut andprojection. Under somewhat standard assumptions, we show that the naturalcomplexity function has polynomial growth. We compute its exponent \alpha interms of the ranks of certain groups which appear in the construction. We givebounds for \alpha. These computations apply to some well known tilings, such asthe octagonal tilings, or tilings associated with billiard sequences. A link ismade between the exponent of the complexity, and the fact that the cohomologyof the associated tiling space is finitely generated over \Q. We show that sucha link cannot be established for more general tilings, and we present acounter-example in dimension one.

Complexity and cohomology for cut-and-projection tilings