Homology of free quantum groups

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 ${\mathrm{\ell }}^2$-homology of $A_o(n)$ and estimates on the free entropy dimension of its set of generators. In particular, we show that the ${\mathrm{\ell }}^2$ Betti-numbers of $A_o(n)$ all vanish and that the free entropy dimension is less than 1.