2023/03/13 by Sam Nariman, Nariman, Sam, Mehdi Yazdi +1
Mathematics · #20F60 #22A22 #43A07 #57R30 #58H05 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2303.07443
openalex publication_date 2023/03/13 · openalex created_date 2023/03/16 · openalex updated_date 2026/07/28
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 H1(L;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 π1(L) is amenable and H1(L;ℝ)=0, then for every transversely orientable codimension-one foliation F having L as a leaf, there is a neighborhood of F in the space of C1,0 foliations with Epstein C0 topology consisting entirely of foliations that are locally a product L × ℝ.