Pan–Rong conjecture in dimension six

Prove that the fundamental group of every open six-manifold with nonnegative Ricci curvature whose universal cover has Euclidean volume growth is finitely generated, as required by the dimension-six case of the Pan–Rong conjecture.

Background

The main theorem establishes finite generation and virtual abelianness in dimension five. For dimensions at least six, under additional asymptotic-cone splitting assumptions, the paper proves only a quantitative virtual-abelianness statement: the fundamental group contains an abelian subgroup of bounded index. Thus the dimension-six finite-generation conclusion of the original conjecture is not established in the paper.