  Let M be a compact manifold. We show the identity component
$\mathrm{Homeo}_0(M)$ of the group of self-homeomorphisms of M has a
well-defined quasi-isometry type, and study its large scale geometry. Through
examples, we relate this large scale geometry to both the topology of M and the
dynamics of group actions on M. This gives a rich family of examples of
non-locally compact groups to which one can apply the large-scale methods
developed in previous work of the second author.
