TITLE: On the global structure of the Gromov-Hausdorff metric space
QUESTION [26 upvotes]: This is a purely idle question, which emerged during a conversation with a friend about what is (not) known about the space of compact metric spaces. I originally asked this question at math.stackexchange (https://math.stackexchange.com/questions/1356066/global-structure-of-the-gromov-hausdorff-space), but received no answers even after bountying.
Background. If $A, B$ are compact subsets of a metric space $M$, the Hausdorff distance between $A$ and $B$ is the worst worst best distance between their points: $$d_H(A, B)=\max\{\sup\{\inf\{d(a, b): b\in B\}: a\in A\}, \sup\{\inf\{d(a, b): a\in A\}: b\in B\}\}.$$ For two compact metric spcaes $X, Y$, their Gromov-Hausdorff distance is the infimum (in fact, minimum) over all isometric embeddings of $X, Y$ into $Z$ via $f, g$ of $d_H(f(X), g(Y))$. The Gromov-Hausdorff space $\mathcal{GH}$ is then the "set" of isometry classes of compact metric spaces, with the metric $d_{GH}$.)
Question. How homogeneous is $\mathcal{GH}$? For example: while distinct points in $\mathcal{GH}$ are in fact distinguishable in a broad sense, it's not clear that distinct points can always be distinguished just by the metric structure of $\mathcal{GH}$, so it's reasonable to ask:

Are there any nontrivial self-isometries of $\mathcal{GH}$?

There are of course lots of related questions one can ask (e.g., autohomeomorphisms instead of isometries); I'm happy for any information about the homogeneity of $\mathcal{GH}$.

Please feel free to add tags if more are appropriate.

REPLY [10 votes]: We have written a text with more details on the proof given by George Lowther. Here one can see the result:
https://arxiv.org/pdf/1806.02100.pdf
However, locally there are many self-isometries, see
https://arxiv.org/pdf/1611.04484.pdf