  It was conjectured by Milnor in 1968 that the fundamental group of a complete
manifold with nonnegative Ricci curvature is finitely generated. The main
result of this paper is a counterexample, which provides an example $M^7$ with
${\rm Ric}\geq 0$ such that $\pi_1(M)=\mathbb{Q}/\mathbb{Z}$ is infinitely
generated.
  There are several new points behind the result. The first is a new
topological construction for building manifolds with infinitely generated
fundamental groups, which can be interpreted as a smooth version of the fractal
snowflake. The ability to build such a fractal structure will rely on a very
twisted gluing mechanism. Thus the other new point is a careful analysis of the
mapping class group $\pi_0\text{Diff}(S^3\times S^3)$ and its relationship to
Ricci curvature. In particular, a key point will be to show that the action of
$\pi_0\text{Diff}(S^3\times S^3)$ on the standard metric $g_{S^3\times S^3}$
lives in a path connected component of the space of metrics with ${\rm Ric}>0$.
