  We show that every oriented $n$-dimensional Poincar\'e duality group over a
$*$-ring $R$ is amenable or satisfies a linear homological isoperimetric
inequality in dimension $n-1$. As an application, we prove the Tits alternative
for such groups when $n=2$. We then deduce a new proof of the fact that when
$n=2$ and $R = \mathbb Z$ then the group in question is a surface group.
