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In Fifteen problems about MCG Ivanov stated the following metaconjecture:

Every object naturally associated to a surface S and having a sufficiently rich structure has $Mod(S)$ as its groups of automorphisms. Moreover, this can be proved by a reduction to the theorem about the automorphisms of $C(S)$.

I have found the following examples supporting the above:

1) Torelli buildings and their automorphisms.

2) Automorphisms of the disk complex.

3) Automorphisms of the Complex of Curves.

Q.1) Are there other results supporting this metaconjecture?

Q.2) Are there examples where the result does not uses the techniques of Ivanov's original paper Automorphisms of Complexes of Curves and of Teichmuller Spaces.

Q.3)(Not precise!) What is the meaning of "sufficiently rich structure"?

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EDIT: Brendle-Margalit have released their paper. See here.

One should observe that these are not all examples of Ivanov's metaconjecture (for instance, the automorphism group of the disk complex is the handle body group, not the whole mapping class group). In any case, Dan Margalit and Tara Brendle are currently writing a paper that proves a version of Ivanov's metaconjecture (which contains as special cases almost all the known examples of complexes whose automorphism group are the mapping class group). I saw both of them in Arkansas a couple of weeks ago, and they assured me that the paper would be released soon. Until then, Brendle's lecture at the Arkansas conference was videotaped and I believe that they will (eventually) post a video of it to the conference webpage here. You could also email her -- her talk used slides which were pretty good and I'm sure she would be happy to share them with you.

Brendle-Margalit's theorem is a little complicated to state, so I will only try to give the gist of it. They consider complexes whose elements are connected subsurfaces of a surface whose topology is constrained somehow. For instance, for the curve complex they would look at the complex of all essential annuli in the surface, and for the non separating curve complex they would look at the complex of all non separating annuli in the surface. It turns out that there are obstructions to Ivanov's metaconjecture holding. I don't want to try to describe them all precisely, but the basic idea is that you want to avoid "exchange automorphisms" of the complexes which flip two vertices that can't be flipped by the mapping class group while fixing everything else. An example would be to look at the complex whose vertices are all one-holed tori and all annuli which cut off a one-holed torus. Clearly you can flip a torus $T$ and a regular neighborhood of its boundary curve $\partial T$. One can come up with more complicated variants on this kind of problem, but it turns out that all the trouble comes from these kinds of things.

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Q1. A beautiful survey about these results is the following:

McCarthy-Papadopoulos, Simplicial actions of mapping class groups, in Handbook of Teichmüller theory Volume III

Q2. Luo's proof of the rigidity of the curve complex relies on Grothendieck 's reconstruction principle:

F. Luo, Automorphisms of the curve complex https://arxiv.org/abs/math/9904020

Q3. Brendle and Margalit announced their result more than one year ago but no proof has appeared so far on ArXiv, so I'll give my answer referring only to published (and refereed) literature.

The first counterexample of Ivanov's conjecture was found by McCarthy-Papadopoulos in the paper

McCarthy-Papadopoulos, Automorphisms of the complex of domains, https://arxiv.org/pdf/math/0606592

Given a compact orientable surface $S$ its complex of domains $D(S)$ is a complex whose vertices are isotopy classes of subsurfaces that can be realized disjointly on the surface. The mapping class group $MCG(S)$ acts on it simplicially. If the surface has at least two boundary components, its action is not rigid because there are automorphisms that can exchange a biperipheral pair of pants and the annulus corresponding to its unique essential boundary curve. McCarthy-Papadopoulos actually prove that $$Aut D(S) \cong MCG(S) \rtimes B$$ where $B$ is the subgroup generated by these exchanges. They also point out the following condition, that is very useful to construct other counterexamples of Ivanov's metaconjecture as the one Andy was talking about:

If $K$ is a simplicial complex the following are equivalent:

1) there exists an automorphism $\phi:K\to K$ that exchanges two non-adiacent vertices $v$ and $w$ fixing everything else;
2) $v$ and $w$ have the same link in $K$, that is $\mathrm{Lk}(v, K) = \mathrm{Lk}(w,K)$ as subsets of $K$.

In other words, if the complex is "too rich" then it is not rigid anymore.

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