Infinitesimally rigid Lie foliations with dense leaves

TL;DR

The authors address infinitesimal rigidity of foliations by examining the deformation cohomology $H^{1}(\mathcal{F},N\mathcal{F})$ and focusing on Lie foliations with dense leaves. They develop a framework showing that rigidity imposes strong algebraic constraints on the transverse Lie algebra $\mathfrak{g}$ and the fundamental group, and they introduce reduction techniques enabling passage to simple factors. The central construction uses suspension foliation techniques together with Kazhdan’s property (T) to force vanishing group cohomology, producing new infinitesimally rigid Lie foliations with dense leaves that are not Hausdorff. They provide explicit arithmetic lattice constructions yielding finitely presented, dense subgroups with property (T) and show how these give concrete examples for any compact semisimple $\mathfrak{g}$ without a simple $\mathfrak{so}(3)$ factor. Overall, the paper delivers the first known infinitesimally rigid Riemannian foliations with dense leaves and offers an explicit algorithmic pathway to construct them from arithmetic data.

Abstract

We call a foliation $\mathcal{F}$ on a compact manifold infinitesimally rigid if its deformation cohomology $H^{1}(\mathcal{F},N\mathcal{F})$ vanishes. This paper studies infinitesimal rigidity for a distinguished class of Riemannian foliations, namely Lie foliations with dense leaves. We construct infinitesimally rigid Lie foliations with dense leaves, modeled on any compact semisimple Lie algebra with simple ideals different from $\mathfrak{so}(3)$. To our knowledge, these are the first examples of infinitesimally rigid Riemannian foliations that are not Hausdorff.

Theorems & Definitions (42)

• Lemma 2.1: Heitsch
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Proposition 3.1
• proof : Proof of Prop. \ref{['prop:obstruction']}
• Corollary 3.2
• proof
• Corollary 3.3
• proof