Convergence in this topology is a crucial ingredient of hyperbolic Dehn surgery, a fundamental tool in the theory of hyperbolic 3-manifolds.
Gromov's topology utilizes the Gromov-Hausdorff metric and is defined on pointed hyperbolic 3-manifolds.
This results in the same notion of convergence as above as the thick part is always connected; thus, a large ball will eventually encompass all of the thick part.
As a further refinement, Gromov's metric can also be defined on framed hyperbolic 3-manifolds.
This gives nothing new but this space can be explicitly identified with torsion-free Kleinian groups with the Chabauty topology.