A vanishing theorem for the homology of discrete subgroups of \(\text{Sp}(n,1)\) and \(\text{F}_{4}^{-20}\) (Q2835324)

Let \(G\) be a simple Lie group with rank\(_\mathbb{R}(G) =1\), namely \(\text{SO}(n,1), \text{SU}(n,1),\text{Sp}(n,1), \text{ or F}_4^{-20}\). These groups are, up to isogeny, the groups of orientation preserving isometries of the real, complex, and quaternionic-hyperbolic spaces \(\mathbf{H}^n_\mathbb{R}, \mathbf{H}^n_\mathbb{C}, \mathbf{H}^n_\mathbb{H}\) and the Cayley-hyperbolic plane \(\mathbf{H}^2_\mathbb{O}\), respectively. If \(\mathbb{K}=\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}\), then each \(\mathbf{H}^n_\mathbb{K}\) is a connected, contractible, negatively curved, symmetric Riemannian manifold of dimension \(dn\) where \(d=1,2,4\) or \(8\), with normalized sectional curvatures between \(-4\) and \(-1\). For a discrete torsion-free subgroup \(\Gamma<G\), we have the corresponding locally symmetric manifold \(M_\Gamma=\mathbf{H}^n_\mathbb{K}/\Gamma\).NEWLINENEWLINEThe main result of the present paper is the following homology vanishing theorem: Theorem 1.1. Let \(\Gamma\) be any discrete, finitely generated, torsion-free subgroup of \(\text{Sp}(n,1)\) or \(\text{F}_4^{-20}\). Assume that Vol\((\mathbf{H}^n_\mathbb{K}/\Gamma)= \infty\) and that \(\Gamma\) has no parabolic elements. (a) If \(\Gamma < \text{Sp}(n,1)\) and \(n\geq 2\), then, for all \(\Gamma\)-modules \(V\), \(H_{4n-1}(\Gamma;V)=0\). (b) If \(\Gamma < \text{F}_4^{-20}\), then, for all \(\Gamma\)-modules \(V\), \(H_{13}(\Gamma,V)=H_{14}(\Gamma,V)=H_{15}(\Gamma; V) =0\).NEWLINENEWLINEThe statement of Theorem 1.1. also holds when \(\Gamma\) is cocompact and \(V\) is finite dimensional. An immediate consequence of Theorem 1.1 is the following result concerning the homological dimension gap. Corollary 1.2. If \(\Gamma < \text{Sp}(n,1)\) is any discrete, finitely generated, torsion free subgroup with no parabolic elements, then either the homological dimension hd\((\Gamma) =4n\) or hd\((\Gamma)\leq 4n-2\). Moreover, the value \(4n\) is obtained precisely when \(\Gamma\) is a lattice. Similarly, if \(\Gamma< F_4^{-20}\) satisfies the same condition, then either hd\((\Gamma)=16\) or hd\((\Gamma)\leq 12\). Moreover, the value 16 is obtained precisely when \(\Gamma\) is a lattice. The proof of Theorem 1.1. uses \(\mathbb{K}\)-hyperbolic geometry. The main steps in the proof are: Step 1. Contracting map, Step 2. Arbitrarily small \((4n-1)\)-mass, Step 3. Gromov's principle.