Intersection of transverse foliations in 3-manifolds: Hausdorff leafspace implies leafwise quasi-geodesic

TL;DR

The paper proves that for two transverse foliations $\mathcal{F}_1,\mathcal{F}_2$ with Gromov hyperbolic leaves on a closed 3-manifold $M$ (with non-solvable $\pi_1(M)$), the intersection foliation $\mathcal{G}=\mathcal{F}_1\cap\mathcal{F}_2$ is leafwise quasigeodesic in the leaves of $\widetilde{\mathcal{F}}_1$ and $\widetilde{\mathcal{F}}_2$ if and only if, for every leaf $L$, the induced foliation $\mathcal{G}_L$ has Hausdorff leaf space. The authors develop a multi-step strategy—landing of rays, small visual measure, push-through, and collapsing—to connect topological properties of leaf spaces to geometric control of leaves, with significant implications for partially hyperbolic diffeomorphisms and Anosov flows. They also analyze the role of parabolic leaves and provide an explicit example illustrating the indispensable non-solvable fundamental group hypothesis. The results yield a robust framework for recognizing when intersected foliations exhibit uniform quasigeodesic behavior and how this behavior constrains the ambient dynamics on $M$.$

Abstract

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be transverse two dimensional foliations with Gromov hyperbolic leaves in a closed 3-manifold $M$ whose fundamental group is not solvable, and let $\mathcal{G}$ be the one dimensional foliation obtained by intersection. We show that $\mathcal{G}$ is \emph{leafwise quasigeodesic} in $\mathcal{F}_1$ and $\mathcal{F}_2$ if and only if the foliation $\mathcal{G}_L$ induced by $\mathcal{G}$ in the universal cover $L$ of any leaf of $\mathcal{F}_1$ or $\mathcal{F}_2$ has Hausdorff leaf space. We end up with a discussion on the hypothesis of Gromov hyperbolicity of the leaves.

Figures (9)

• Figure 1: When a leaf $L$ of $\widetilde{\mathcal{F}}_1$ intersects a leaf $E$ of $\widetilde{\mathcal{F}}_2$ in two connected components, the leafspace of $\mathcal{G}_L$ is not Hausdorff.
• Figure 2: Some leaves of $\mathcal{G}_L$ inside $L$ with different landing behaviour. Note that $\mathcal{G}_L$ does not have Hausdorff leaf space because there are three leaves of $\mathcal{G}_L$ which land and define disjoint intervals in $S^1(L)$.
• Figure 3: Depiction of some of the objects appearing in the definition of the neighborhood $V_n(\xi)$. The inner circle represents the unit tangent vectors to $p$ and painted in red are the vectors in $A_p$.
• Figure 4: In the left a piece of leaf $E$ of $\widetilde{\mathcal{F}}_2$ intersecting two leaves $L_0$ and $L_{t_1}$ of $\widetilde{\mathcal{F}}_1$. $E$ intersects $L_0$ in a compact arc $c_0$, and $E$ intersects $L_{t_1}$ in two rays that go to infinity. In the right the leaf of $E$ is depicted and showed how this structure of intersection forces the arcs to split and induce non Hausdorff leaf space.
• Figure 5: A depiction of the sets $H_n$ inside $L$.

Theorems & Definitions (106)

• Theorem 1.1
• Corollary 1.2
• Theorem 2.1
• Remark 2.2
• Theorem 2.3
• Proposition 2.4
• Lemma 2.5
• proof
• proof : Proof of Proposition \ref{['prop-Hsdff2Dand3D']}
• Proposition 2.6