  Let $M$ be an open $n$-manifold with nonnegative Ricci curvature. We prove
that if its escape rate is not $1/2$ and its Riemannian universal cover is
conic at infinity, that is, every asymptotic cone $(Y,y)$ of the universal
cover is a metric cone with vertex $y$, then $\pi_1(M)$ contains an abelian
subgroup of finite index. If in addition the universal cover has Euclidean
volume growth of constant at least $L$, we can further bound the index by a
constant $C(n,L)$.
