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Margulis Lemma on $\text{RCD}(K,N)$ spaces

Qin Deng, Jaime Santos-Rodríguez, Sergio Zamora, Xinrui Zhao

TL;DR

The paper extends the Margulis Lemma to $ ext{RCD}^{\ast}(K,N)$ spaces, including collapsed cases, by developing sharp regularity theory for Regular Lagrangian Flows and combining it with an equivariant Gromov–Hausdorff framework. The core technical advance is a robust induction scheme (via the Induction Theorem) that controls nilpotent structures in fundamental groups along GH limits, aided by GS maps and a re-scaling analysis. The results yield a finite-index, nilpotent-structured subgroup of the fundamental group with bounds depending only on $(K,N)$ and the space's rectifiable dimension, mirroring smooth manifold results and extending them to nonsmooth synthetic-geometry settings. This has significant implications for the topology of $ ext{RCD}^{\ast}(K,N)$ spaces, including collapsed limits, and provides tools to analyze the fundamental group and covering spaces in synthetic Ricci-curvature contexts.

Abstract

We extend the Margulis Lemma for manifolds with lower Ricci curvature bounds to the $\text{RCD}(K,N)$ setting. As one of our main tools, we obtain improved regularity estimates for Regular Langrangian flows on these spaces.

Margulis Lemma on $\text{RCD}(K,N)$ spaces

TL;DR

The paper extends the Margulis Lemma to spaces, including collapsed cases, by developing sharp regularity theory for Regular Lagrangian Flows and combining it with an equivariant Gromov–Hausdorff framework. The core technical advance is a robust induction scheme (via the Induction Theorem) that controls nilpotent structures in fundamental groups along GH limits, aided by GS maps and a re-scaling analysis. The results yield a finite-index, nilpotent-structured subgroup of the fundamental group with bounds depending only on and the space's rectifiable dimension, mirroring smooth manifold results and extending them to nonsmooth synthetic-geometry settings. This has significant implications for the topology of spaces, including collapsed limits, and provides tools to analyze the fundamental group and covering spaces in synthetic Ricci-curvature contexts.

Abstract

We extend the Margulis Lemma for manifolds with lower Ricci curvature bounds to the setting. As one of our main tools, we obtain improved regularity estimates for Regular Langrangian flows on these spaces.
Paper Structure (20 sections, 60 theorems, 233 equations)

This paper contains 20 sections, 60 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Main regularity estimates on RLFs
  3. Induction Theorem
  4. Structure of the paper
  5. Preamble
  6. Notation
  7. $\text{RCD}^{\ast} (K,N)$ spaces; doubling, isometries, covers, and geodesics
  8. Gromov--Hausdorff topology
  9. Equivariant Gromov--Hausdorff convergence
  10. $\delta$-splittings
  11. Regular Lagrangian Flows
  12. Group norms
  13. Group Theory
  14. Groups of connected components
  15. Proof of main regularity estimates on RLFs
  16. ...and 5 more sections

Key Result

Theorem 1.1

For each $K \in \mathbb{R}$, $N \geq 1$, there exist $\varepsilon > 0$ and $C \in \mathbb{N}$ such that if $(X,d, \mathfrak{m} ,p)$ is a pointed $\text{RCD}^{\ast} (K , N)$ space of rectifiable dimension $n$, the image of the map induced by the inclusion contains a subgroup of index $\leq C$ that admits a nilpotent basis of length $\leq n$.

Theorems & Definitions (117)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.4
  • Corollary 1.5
  • Definition 1.6
  • Example 1.7
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • Theorem 1.11
  • ...and 107 more