Symmetries of Gromov-Hausdorff Distance

The famous Gromov-Hausdorff distance measures the similarity degree of metric spaces. Since it both satisfies the triangle inequality and vanishes for isometric spaces, it induces correctly a correspondent distance on isometry classes of metric spaces. The collection of all such classes form a proper class in terms of Von Neumann–Bernays–Gödel set theory. We call such proper class by Gromov-Hausdorff class and denote it as GH. The main question for the talk is to discuss what are the isometric mappings (local and global) of GH. One of the most investigated part of GH is the Gromov-Hausdorff space M consisting of all non-empty compact metric spaces (considered up to isometry).
We start with a sketch of Ivanov-Tuzhilin's correction of the "George Lowther" proof (perhaps "George Lowther" is a pseudonym) that the isometry group of M is trivial. Then we discuss some local isometries of M: it turns out that there are a lot of them. At last, we formulate some conjectures concerning the whole Gromov-Hausdorff class GH.