  Let $(M^{n},g)$ be a compact Riemannian manifold with $Ric\geq-(n-1) $. It is
well known that the bottom of spectrum $\lambda_{0}$ of its unverversal
covering satisfies $\lambda_{0}\leq(n-1) ^{2}/4 $. We prove that equality holds
iff $M$ is hyperbolic. This follows from a sharp estimate for the Kaimanovich
entropy.
