Let G be a torsion-free, discrete subgroup of either the isometry group of quaternionic or Cayley hyperbolic space; that is, up to isogeny, $G < Sp(n;1)$ for $n \geq 2$ or $G < F4^{-20}$. We prove that if G contains no parabolics then there is a gap in the possible homological dimension $hd(G)$ of $G$. Namely, if $G < Sp(n;1)$, either $hd(G) = 4n$ or $hd(G)\leq 4n-2$, and if $G < F4^{-20}$, then either $hd(G) = 16$ or $hd(G)\leq 12$. This result does not hold in the real or complex hyperbolic cases, or if $G$ is allowed to have parabolics (even for subgroups of lattices). Our method requires a generalization of work of Besson–Courtois–Gallot on estimates of p–Jacobians of natural maps. We also generalize an inequality of M. Kapovich between the homological dimension and critical exponent for discrete subgroups of the isometry group of real hyperbolic n–space. This is joint work with Benson Farb and Ben McReynolds.
