We present some local topological rigidity results for the set S of
non-geometric, compact -- with possibly empty boundary and no
spherical boundary components --, orientable Riemannian 3-manifolds
having torsionfree fundamental group, with bounded entropy and
diameter. By "local", we mean that we consider S endowed with the
Gromov-Hausdorff-topology. We shall provide examples to show the
necessity of the assumptions and discuss some open problems.
Moreover, we shall give a proof of the finiteness of the homeomorphism
types of the manifolds in S. These are joint works with A. Sambusetti
(Rome, Sapienza).