Distinguishing 4-dimensional geometries via profinite completions

Abstract

It is well-known that there are 19 classes of geometries for 4-dimensional manifolds in the sense of Thurston. We could ask that to what extent the geometric information is revealed by the profinite completion of the fundamental group of a closed smooth geometric 4-manifold. In this paper, we show that if two geometric manifolds, neither of whose geometries is $$\mathbb {H}^{4}$$ or $$\mathbb {H}^{2}_{\mathbb {C}}$$ , share the same profinite completion then they have the same geometry. Moreover, despite the fact that not every smooth 4-manifold could admit one geometry in the sense of Thurston, some 4-dimensional manifolds with Seifert fibred structures are indeed geometric. For a closed orientable Seifert fibred 4-manifold M, we show that whether M is geometric could be detected by the profinite completion of its fundamental group.