  We compute the Hochschild homology of the free orthogonal quantum group
$A_o(n)$. We show that it satisfies Poincar\'e duality and should be considered
to be a 3-dimensional object. We then use recent results of R. Vergnioux to
derive results about the $\ell^2$-homology of $A_o(n)$ and estimates on the
free entropy dimension of its set of generators. In particular, we show that
the $\ell^2$ Betti-numbers of $A_o(n)$ all vanish and that the free entropy
dimension is less than 1.
