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On $C^0$-stability of compact leaves with amenable fundamental group

Sam Nariman, Mehdi Yazdi

Abstract

In his work on the generalization of the Reeb stability theorem, Thurston conjectured that if the fundamental group of a compact leaf $L$ in a codimension-one transversely orientable foliation is amenable and if the first cohomology group $H^1(L;\mathbb{R})$ is trivial, then $L$ has a neighborhood foliated as a product. This was later proved as a consequence of Witte-Morris' theorem on the local indicability of amenable left orderable groups and Navas' theorem on the left orderability of the group of germs of orientation-preserving homeomorphisms of the real line at the origin. In this note, we prove that Thurston's conjecture also holds for any foliation that is sufficiently close to the original foliation. Hence, if the fundamental group $π_1(L)$ is amenable and $H^1(L;\mathbb{R})=0$, then for every transversely orientable codimension-one foliation $\mathcal{F}$ having $L$ as a leaf, there is a neighborhood of $\mathcal{F}$ in the space of $C^{1,0}$ foliations with Epstein $C^0$ topology consisting entirely of foliations that are locally a product $L \times \mathbb{R}$.

On $C^0$-stability of compact leaves with amenable fundamental group

Abstract

In his work on the generalization of the Reeb stability theorem, Thurston conjectured that if the fundamental group of a compact leaf in a codimension-one transversely orientable foliation is amenable and if the first cohomology group is trivial, then has a neighborhood foliated as a product. This was later proved as a consequence of Witte-Morris' theorem on the local indicability of amenable left orderable groups and Navas' theorem on the left orderability of the group of germs of orientation-preserving homeomorphisms of the real line at the origin. In this note, we prove that Thurston's conjecture also holds for any foliation that is sufficiently close to the original foliation. Hence, if the fundamental group is amenable and , then for every transversely orientable codimension-one foliation having as a leaf, there is a neighborhood of in the space of foliations with Epstein topology consisting entirely of foliations that are locally a product .
Paper Structure (9 sections, 7 theorems, 30 equations)

This paper contains 9 sections, 7 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Left-invariant orders on groups
  3. Deformations of foliations and their holonomies
  4. Étale groupoids
  5. Fundamental groupoid of a foliation
  6. Epstein topology on the space of foliations
  7. Deformation of a foliation
  8. Proof of Theorem \ref{['thm:main']}
  9. Questions

Key Result

Theorem 1.1

Let $M$ be a compact connected smooth manifold. Let $\mathcal{F}$ be a transversely oriented codimension-one $C^{1,0}$ foliation on $M$ with a compact leaf $L$ such that $\pi_1(L)$ is amenable and $H^1(L;\mathbb{R}) = 0$. Then any $C^{1,0}$ foliation that is $C^0$ close to $\mathcal{F}$ (i.e. in Eps

Theorems & Definitions (20)

  • Theorem 1.1
  • Example 2.1
  • Theorem 2.2: Dynamic realization
  • proof
  • Remark 2.3
  • Theorem 2.4
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • proof
  • ...and 10 more