  For any discrete, torsion-free subgroup $\Gamma$ of $\mathrm{Sp}(n,1)$
(resp.\ $\mathrm{F}_4^{-20}$) with no parabolic elements, we prove that
$H_{4n-1}(\Gamma;V)=0$ (resp.\ $H_i(\Gamma;V)=0$ for $i=13,14,15$) for any
$\Gamma$--module $V$. The main technical advance is a new bound on the
$p$--Jacobian of the barycenter map of Besson--Courtois--Gallot. We also apply
this estimate to obtain an inequality between the critical exponent and
homological dimension of $\Gamma$, improving on work of M.~Kapovich.
