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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 applythis estimate to obtain an inequality between the critical exponent andhomological dimension of $\Gamma$, improving on work of M.~Kapovich.

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