We will investigate the local topological rigidity fenomenon for 3 dimensional Riemannian manifolds under restraints of the volume entropy. A Riemannian manifold M is said to be locally topologically rigid if any other manifold M' sufficiently close to M with respect to the Gromov-Hausdorff distance is diffeomorphic to M. This property is too strong to hold in the class of Riemannian manifolds without further assumptions, hence we restrict ourselves to the class of Riemannian 3-manifolds whose volume entropy is bounded from above.
Local topological rigidity has already been investigated by Cheeger and Colding, who proved it for Riemannian manifolds of any dimension with a uniform lower bound on the Ricci curvature.
The volume entropy can be thought as a weak, large scale, version of the curvature, hence the upper bound on the entropy is a natural weakening of the (negative) lower bound on the Ricci curvature, this motivates the assumption of the upper bound on the entropy. In this setting only large scale information is provided by the assumption so one needs to recover topological information on a small scale by other means. This is the reason why we restrict to 3 dimensional manifolds.
In this setting partial results are already known: in a recent paper due to Cerocchi and Sambusetti the local topological rigidity is shown to hold for torsionless non-geometric Riemannian 3-manifolds under the assumption of the volume entropy upper bound.
For the other eight geometric cases instead the question is still open, and to be investigated in this project.
In particular we will prove that local topological rigidity holds for torsionless 3-manifolds of hyperbolic type, due to their very rich geometry. On the other hand, for less negatively curved geometries, we will try to recover weaker results, or the same result under stronger assumptions. It is an interesting open question that will be explored to determine which hypothesis are the most adapt to work in this direction.
This research is placed along a well defined path of generalising classical results in Riemannian geometry to a weaker setting, and try to answer a question that was left open in a recent article of Cerocchi and Sambusetti: "Local topological rigidity for non-geometric 3 manifolds" where they obtained a version of Colding's stability for non-geometric, torsionless, Riemannian 3-manifolds, again weakening the classical assumption of the lower bound of the Ricci curvature to an upper on the entropy.
As pointed out in their paper, while these facts are in the same spirit of Kapovich and Leeb quasi-isometric (virtual)
rigidity results for the fundamental group of non-geometric 3-manifolds, they are completely different. In fact, beside drawing stronger conclusions (as the fundamental group cannot be determined from the quasi-isometry type alone), without any assumption on the entropy, one can easily produce non-geometric manifolds X, X0 which are arbitrarily close in the Gromov-Hausdorff distance and which do not have quasi-isometric fundamental groups.
Along this path lies also a recent paper, making use of a celebrated paper of Breuillard, Green, and Tao, Besson, Curtois, Gallot, and Sambusetti proved a version of the classical Bishop-Gromov inequality, and of the Margulis lemma (which classically require respectively a lower bound on the Ricci curvature and an upper bound on the sectional curvature) for Gromov hyperbolic spaces under the assumption of bounded entropy.
As already explained in the methodology section of the previous part, Cerocchi and Sambusetti are able to prove an entropy-systolic inequality thanks to the presence of enough free subgroups in the fundamental group, and then deduce their results. For one of the eight geometries, namely manifolds of hyperbolic type, a similar approach is promising, since their fundamental groups, being Gromov-hyperbolic and hence of hyperbolic growth, contain many free subgroups, just as in the non-geometric case. In this case our knowledge of hyperbolic groups, and fundamental groups of 3-manifolds is indeed enough to obtain our results, leading to a sure improvement of the state of the art.
Concerning the other geometries, while the exponential volume entropy is not as strong of an invariant as in the hyperbolic type case, the knowledge about the corresponding spaces is much more detailed. In particular they are Seifert spaces, which are classified completely by their well known fundamental group (a part for a couple of exceptional cases). This will make it much easier to work out computations in these cases in order to get interesting examples and results.
Finally, regarding the applications, we shall note that the information regarding the local topological rigidity of spaces could be interesting even if it is known only for the class of RIemannian 3-manifolds. In fact some real world structures of great interest show geometric (in the sense of Thurston) behaviour: for example the densely packed DNA filaments in the cells named chromatin arrange themselves as a geometric 3-manifold (with boundary) corresponding to the twisted line bundle over the sphere. Applications in this direction look extremely interesting while currently unexplored to the proposer.