  Following Thurston's geometrisation picture in dimension three, we study
geometric manifolds in a more general setting in arbitrary dimensions, with
respect to the following problems: (i) The existence of maps of non-zero degree
(domination relation or Gromov's order); (ii) The Gromov-Thurston monotonicity
problem for numerical homotopy invariants with respect to the domination
relation; (iii) The existence of Anosov diffeomorphisms (Anosov-Smale
conjecture).
