A vanishing theorem for the homology of discrete subgroups of $\mathrm{Sp}(n,1)$ and $\mathrm{F}_4^{-20}$

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.