On Riemannian manifold with Ricci curvature bounded below
(For example, flat torus. It has ${\bf Z}^2$ as fundamental group, which is nilpotent (=almost abelian). In fact Ricci curvature bouned below-condition is strong. For example, fundamental group contains a nilpotent group of finite index if its diameter is smaller than $C(n)$, and it is a conjecture that a closed manifold with positive isotropic curvature has virtually free fundamental group.
And I do not know how almost flat manifolds are related with manifolds with Ricci curvature bounded below. Here almost flat is the condition ${\rm sec}(x) d(x,x_0)^2 \leq \epsilon$. For example unit sphere is not almost flat. Gromov and Ruh have shown that an almost flat manifold is in fact an infranilmanifold. And if $M$ is compact then it has a nilpotent fundamental group, and is a nilmanifold.),
if the fundamental group has torsion which is abelian then is there some good topological invariant ? If a closed manifold is positively curved then it has a lot of symmetries, for instance a lens space (cf. Grove and Searle).
And if you has some knowledge about how a torsion influence the topology on Riemannian manifolds, tell me, please.