Poincaré inequality on complete Riemannian manifolds with Ricci curvature bounded below

Abstract: We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincar\'e inequalities. A global, uniform Poincar\'e inequality for horospheres in the universal cover of a closed, $n$-dimensional Riemannian manifold with pinched negative sectional curvature follows as a corollary.