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Nonnegative Ricci curvature, nilpotency, and Hausdorff dimension

Jiayin Pan

TL;DR

The paper examines open manifolds with nonnegative Ricci curvature and escape rate not equal to 1/2, proving that the nilpotency step of a finite-index torsion-free nilpotent subgroup of the fundamental group is detected in the asymptotic geometry of the universal cover. Using equivariant Gromov–Hausdorff convergence, distance-gap arguments via tunnels, and the construction of adapted bases/maps, it shows that for any equivariant asymptotic cone, the limit orbit Ly carries a simply connected nilpotent Lie-group structure with step at most that of the original subgroup, and that there exists a cone where dim_H(Ly) ≥ the nilpotency step. Consequently, the nilpotency step yields lower bounds on the Hausdorff dimension of limit orbits, and in favorable cases (e.g., low asymptotic dimension) implies virtual abelianness of π1(M). The results unify and extend prior work on metric cones at infinity and the Grushin-type asymptotic cones, providing a robust conduit from algebraic group properties to the asymptotic metric geometry of Ricci-nonnegative manifolds. These developments offer a principled route to bound or determine virtual properties of π1(M) from large-scale geometric data.

Abstract

Let $M$ be an open (complete and non-compact) manifold with $\mathrm{Ric}\ge 0$ and escape rate not $1/2$. It is known that under these conditions, the fundamental group $π_1(M)$ has a finitely generated torsion-free nilpotent subgroup $\mathcal{N}$ of finite index, as long as $π_1(M)$ is an infinite group. We show that the nilpotency step of $\mathcal{N}$ must be reflected in the asymptotic geometry of the universal cover $\widetilde{M}$, in terms of the Hausdorff dimension of an isometric $\mathbb{R}$-orbit: there exist an asymptotic cone $(Y,y)$ of $\widetilde{M}$ and a closed $\mathbb{R}$-subgroup $L$ of the isometry group of $Y$ such that its orbit $Ly$ has Hausdorff dimension at least the nilpotency step of $\mathcal{N}$. This resolves a question raised by Wei and the author.

Nonnegative Ricci curvature, nilpotency, and Hausdorff dimension

TL;DR

The paper examines open manifolds with nonnegative Ricci curvature and escape rate not equal to 1/2, proving that the nilpotency step of a finite-index torsion-free nilpotent subgroup of the fundamental group is detected in the asymptotic geometry of the universal cover. Using equivariant Gromov–Hausdorff convergence, distance-gap arguments via tunnels, and the construction of adapted bases/maps, it shows that for any equivariant asymptotic cone, the limit orbit Ly carries a simply connected nilpotent Lie-group structure with step at most that of the original subgroup, and that there exists a cone where dim_H(Ly) ≥ the nilpotency step. Consequently, the nilpotency step yields lower bounds on the Hausdorff dimension of limit orbits, and in favorable cases (e.g., low asymptotic dimension) implies virtual abelianness of π1(M). The results unify and extend prior work on metric cones at infinity and the Grushin-type asymptotic cones, providing a robust conduit from algebraic group properties to the asymptotic metric geometry of Ricci-nonnegative manifolds. These developments offer a principled route to bound or determine virtual properties of π1(M) from large-scale geometric data.

Abstract

Let be an open (complete and non-compact) manifold with and escape rate not . It is known that under these conditions, the fundamental group has a finitely generated torsion-free nilpotent subgroup of finite index, as long as is an infinite group. We show that the nilpotency step of must be reflected in the asymptotic geometry of the universal cover , in terms of the Hausdorff dimension of an isometric -orbit: there exist an asymptotic cone of and a closed -subgroup of the isometry group of such that its orbit has Hausdorff dimension at least the nilpotency step of . This resolves a question raised by Wei and the author.
Paper Structure (20 sections, 48 theorems, 289 equations)

This paper contains 20 sections, 48 theorems, 289 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivariant Gromov-Hausdorff convergence
  4. Nilpotent groups
  5. Escape rate
  6. Motivating examples
  7. Equivariant Gromov-Hausdorff distance gaps
  8. Tunnels
  9. Statements of distance gaps and a rough idea of the proof
  10. Adapted bases and adapted maps
  11. Proof of the distance gaps
  12. Asymptotic orbits
  13. Uniform type of asymptotic orbits and proof of Theorem A(1)
  14. One-parameter orbits
  15. Proof of Proposition C(1)
  16. ...and 5 more sections

Key Result

Theorem 2.2

FuFY Let be a Gromov-Hausdorff convergence sequence and let $G_i$ be closed subgroups of $\mathrm{Isom}(Y_i)$. Then the followings hold. (1) After passing to a subsequence, we have an equivariant Gromov-Hausdorff convergence sequence where $H$ is closed subgroup of $\mathrm{Isom}(Z)$. (2) The sequence of quotient spaces converges

Theorems & Definitions (108)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Definition 2.8
  • Proposition 2.9
  • Corollary 2.10
  • ...and 98 more