Examples of Ricci limit spaces with non-integer Hausdorff dimension

Abstract

Let M be an open (complete and non-compact) manifold with Ric ≥ 0 and escape rate not 1/2. It is known that under these conditions, the fundamental group π1(M ) has a finitely generated torsion-free nilpotent subgroup N of finite index, as long as π1(M ) is an infinite group. We show that the nilpotency step of N must be reflected in the asymptotic geometry of the universal cover M , in terms of the Hausdorff dimension of an isometric Rorbit: there exist an asymptotic cone (Y, y) of M and a closed R-subgroup L of the isometry group of Y such that its orbit Ly has Hausdorff dimension at least the nilpotency step of N . This resolves a question raised by Wei and the author (see [24, Remark 1.7] and [23, Conjecture 0.2]) and extends previous results on virtual abelianness by the author [21,22,23].

One dimensional RCD spaces always satisfy the regular Weyl's law

“…The coincident of these two notion of dimension was open. However, recently Pan and Wei proved that there exists an RCD(K, N ) space whose Hausdorff dimension is strictly larger than essential one ( [19]).…”

The Grushin hemisphere as a Ricci limit space with curvature ≥1

“…Comparing Theorem A with Pan-Wei's construction [9], one of the main differences here is the positive Ricci curvature lower bound. In [9], compact Ricci limit spaces with large Hausdorff dimension were constructed, but they must have negative Ricci curvature somewhere.…”

The Grushin hemisphere as a Ricci limit space with curvature $\ge 1$

“…Comparing Theorem A with Pan-Wei's construction [8], one of the main differences here is the positive Ricci curvature lower bound. In [8], compact Ricci limit spaces with large Hausdorff dimension were constructed, but they must have negative Ricci curvature somewhere.…”

“…Comparing Theorem A with Pan-Wei's construction [8], one of the main differences here is the positive Ricci curvature lower bound. In [8], compact Ricci limit spaces with large Hausdorff dimension were constructed, but they must have negative Ricci curvature somewhere. On a technical note, we remark that the fundamental group plays an essential role in Pan-Wei's construction, while in Theorem A we directly construct the metrics on a sphere.…”

“…The structure of Ricci limit spaces is crucial in understanding Ricci curvature and has been studied extensively since the seminal works by Cheeger and Colding. In a previous joint work with Wei [8], we have constructed a Ricci limit space, as the asymptotic cone of a complete Riemannian metric on R n+1 with Ric > 0, such that its Hausdorff dimension exceeds its rectifiable dimension. The limit space is a halfplane [0, ∞) × R; its singular set is the boundary {0} × R with Hausdorff dimension 1 + α, where α > 0 can be any prior chosen number.…”