The Fundamental Groups And Discretizations Of Manifolds With Integral Ricci Curvature Bounds

In this paper, by making use of the improved volume comparison lemmas established by P. Petersen and G. Wei [12], we study the fundamental groups and discretizations of some manifolds. The main results consists of two parts, in the first part, we show that every finitely generated subgroup of the fundamental group of an n-manifold M with integral Ricci curvature bounds is of polynomial growth with degree ≤ n (or ≤ 2p). This generalizes Theorem 1 of paper [11]. In the second part, we generalize the rough isometry between the noncompact Riemannian manifold with Ricci curvature bounded uniformly from below and any of its discretizations to that of the noncompact Riemannian manifold with integral Ricci curvature bounds.