Open manifolds with asymptotically nonnegative Ricci curvature and large   volume growth
  In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature and large volume growth. We prove that they have finite topological types under some curvature decay and volume growth conditions. We also generize it to the manifolds with $k$-th asymptotically nonnegative Ricci curvature by using extensions of Abresch-Gromoll's excess function estimate. 
Yuntao Zhang
math.DG