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I hope the questions are not too vague.

  1. Is the mapping class group of an orientable punctured surface $CAT(0)$ ?

  2. Is any of the remarkable simplicial complexes (curve complex, arc complex...) built on a punctured surface $CAT(0)$?

  3. Is there any "nice" action (say, proper or cocompact) of the mapping class group on a $CAT(0)$ space?

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(1) Bridson showed that if a mapping class group of a surface (of genus at least 3) acts on a CAT(0) space, then Dehn twists act as elliptic or parabolic elements. This implies that the mapping class groups of genus $\geq 3$ are not CAT(0) (Edit: as pointed out by Misha in the comments, this was originally proved by Kapovich and Leeb, based on an observation of Mess that there is a non-product surface-by-$\mathbb{Z}$ subgroup of the mapping class group of a genus $\geq 3$ surface). On the other hand, the mapping class group of a genus 2 surface acts properly on a CAT(0) space (this is not surprising, since it is linear). I think it's unresolved whether the mapping class group of genus 2 is CAT(0) though (this is essentially equivalent to the same question for the 5-strand braid group).

(2) The curve complex cannot admit a CAT(0) metric, since it is homotopy equivalent to a wedge of spheres.

(3) The mapping class group acts cocompactly by isometries on the completion of the Weil-Petersson metric on Teichmuller space, which is CAT(0). However, this metric is not proper (although as Bridson shows above, the action is semisimple, Dehn twists acting by elliptic isometries).

So I guess it's unresolved whether there is a proper action of the mapping class groups of genus $\geq 3$ on a CAT(0) space (where the Dehn twists act as parabolics). This is unsurprising, since it is unknown whether these groups are linear (a finitely generated linear group acts properly on a CAT(0) space which is a product of symmetric spaces and buildings).

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    $\begingroup$ Ian: This is theorem 4.2 of M. Kapovich, B. Leeb, Actions of discrete groups on Hadamard spaces, Math. Annalen, Bd. 306 (1996) p. 341-352. $\endgroup$
    – Misha
    Commented Nov 11, 2013 at 10:28
  • $\begingroup$ @Misha: The published title is a bit different, "Actions of discrete groups on nonpositively curved spaces" $\endgroup$
    – Lee Mosher
    Commented Nov 11, 2013 at 15:59
  • $\begingroup$ @Misha: Right, I thought of this proof (you use the observation of Geoff Mess that there's a non-trivial circle bundle over a surface?), but I thought it was only the case of smooth non-positively curved manifolds. $\endgroup$
    – Ian Agol
    Commented Nov 11, 2013 at 19:20
  • $\begingroup$ Ian: This is a general argument based on product decomposition of parallel sets in CAT(0) spaces. $\endgroup$
    – Misha
    Commented Nov 11, 2013 at 19:31
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    $\begingroup$ Brady and MacCammond showed in arxiv.org/abs/0909.4778 that the 5-strand braid group is CAT(0). $\endgroup$
    – Luc
    Commented Nov 14, 2013 at 13:41

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