  We prove that almost all arc complexes do not admit a CAT(0) metric with
finitely many shapes, in particular any finite-index subgroup of the mapping
class group does not preserve such a metric on the arc complex. We also show
the analogous statement for all but finitely many disc complexes of
handlebodies and free splitting complexes of free groups. The obstruction is
combinatorial. These complexes are all hyperbolic and contractible but despite
this we show that they satisfy no combinatorial isoperimetric inequality: for
any n there is a loop of length 4 that only bounds discs consisting of at least
n triangles.
  On the other hand we show that the curve complexes satisfy a linear
combinatorial isoperimetric inequality, which answers a question of Andrew
Putman.
