  We show that if (M,\omega) is a closed symplectic manifold which admits a
nontrivial Hamiltonian vector field all of whose contractible closed orbits are
constant, then Hofer's metric on the group of Hamiltonian diffeomorphisms of
(M,\omega) has infinite diameter, and indeed admits infinite-dimensional
quasi-isometrically embedded normed vector spaces. A similar conclusion applies
to Hofer's metric on various spaces of Lagrangian submanifolds, including those
Hamiltonian-isotopic to the diagonal in M x M when M satisfies the above
dynamical condition. To prove this, we use the properties of a Floer-theoretic
quantity called the boundary depth, which measures the nontriviality of the
boundary operator on the Floer complex in a way that encodes robust
symplectic-topological information.
