Volume, diameter and the first eigenvalue of locally symmetric spaces of rank one (Q1080197)

Let X be a symmetric space of rank one with compact quotient V. If X is real hyperbolic and dim \(X\geq 4\) or complex hyperbolic then \(diam(V)\leq c_ n vol(V)\) where \(c_ n>0\) depends only on \(n=\dim X\). If X is the hyperbolic space over the quaternions or the octonian plane then: \(diam(V)\leq c_ n+(13/2)\log vol(V)\). In general if X is symmetric of rank one and dim \(X\geq 4\) then \(\lambda_ 1\leq (a_ n+b_ n Log vol(V))/diam(V)\) where \(\lambda_ 1\) is the first nonzero eigenvalue of the Laplacian of V.