The rationality of Sol-manifolds

Let Γ be the fundamental group of a manifold modeled on 3-dimensional Sol geometry. We prove that Γ has a finite index subgroup G which has a rational growth series with respect to a natural generating set. We do this by enumerating G by a regular language. However, in contrast to most earlier proofs of this sort our regular language is not a language of words in the generating set, but rather reflects a different geometric structure in G.