Chris Connell

Let $G$ be a torsion-free, finitely generated discrete subgroup of either the isometry group of quaternionic or Cayley hyperbolic space; that is, up to isogeny, $G < Sp(n;1)$ for $n \ge 2$ or $G < F_4^{-20}$. We prove that if $G$ contains no parabolics then there is a gap in the possible homological dimension $\mathrm{hd}(G)$ of $G$. Namely, if $G < Sp(n;1)$, either $\mathrm{hd}(G) = 4n$ or $\mathrm{hd}(G)\leq 4n-2$, and if $G < F_4^{-20}$, then either $\mathrm{hd}(G) = 16$ or $\mathrm{hd}(G)\leq 12$. 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.