Distinguishing 4-dimensional geometries via profinite completions

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