  A Riemannian manifold is a called a good rational expander in dimension $i$
if every $i$-cycle bounds a rational $i+1$-chain of comparatively small volume.
We construct 3-manifolds which are good expanders in all dimensions. On the
other hand, we show that expanders must be topologically complicated: they must
have lots of torsion homology. We also give some applications to topological
overlap problems, constructing examples of 3-manifolds with large width over
$\mathbb R^2$.
