Classical Notions and Problems in Thurston Geometries

Of the Thurston geometries, those with constant curvature geometries (Euclidean E3, hyperbolic H3, spherical S3) have been extensively studied, but the other five geometries, H2xR, S2xR, Nil, <^>SL2R, Sol have been thoroughly studied only from a differential geometry and topological point of view. However, classical concepts highlighting the beauty and underlying structure of these geometries - such as geodesic curves and spheres, the lattices, the geodesic triangles and their surfaces, their interior sum of angles and similar statements to those known in constant curvature geometries - can be formulated. These have not been the focus of attention. In this survey, we summarize our results on this topic and pose additional open questions.