# Ricci Curvature, Fundamental Group and the Milnor Conjecture

# Ricci Curvature, Fundamental Group and the Milnor Conjecture

It was conjectured by Milnor in 1968 that the fundamental group of a complete manifold with nonnegative Ricci curvature is finitely generated. In this talk we will discuss a counterexample, which provides an example M^7 with Ric>= 0 such that \pi_1(M)=Q/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_0Diff(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_0Diff(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 Ric>0.