According to Wikipedia's article on the geometrization conjecture, one of the eight possible geometric structures for a manifold is the euclidean, or $E^3$. The article on Wikipedia says that this geometry can be modeled as a left invariant metric on one of Bianchi's groups.
Unfortunately, the article has no more information on that, and I am looking for references, preferably on academic journals.
If anybody knows, could please let me know?