Nonnegative Ricci curvature, nilpotency, and Hausdorff dimension

Abstract Let M be an open (complete and non-compact) manifold with Ric 0 Ric ≥ 0 and escape rate not 1/2. It is known that under these conditions, the fundamental group ₁ (M) π 1 (M) has a finitely generated torsion-free nilpotent subgroup N N of finite index, as long as ₁ (M) π 1 (M) is an infinite group. We show that the nilpotency step of N N must be reflected in the asymptotic geometry of the universal cover M M ~, in terms of the Hausdorff dimension of an isometric R R -orbit: there exist an asymptotic cone (Y, y) of M M ~ and a closed R R -subgroup L of the isometry group of Y such that its orbit Ly has Hausdorff dimension at least the nilpotency step of N N. This resolves a question raised by Wei and the author (see Pan and Wei in Geom Funct Anal 32: 676–685, 2022, Remark 1. 7 and Pan in Geom Topol 28: 1409–1436, 2024, Conjecture 0. 2).