Left orderability and taut foliations with orderable cataclysm

The paper establishes that the fundamental group of a closed, orientable, irreducible 3-manifold is left orderable if the manifold admits a co-orientable taut foliation with orderable cataclysm, thereby providing an elementary proof for the left orderability of fundamental groups in the presence of Anosov or pseudo-Anosov flows and extending this result to infinitely many Dehn fillings.

The Gist

The Big Picture: The "L-Space" Mystery

Imagine the universe of 3D shapes (called 3-manifolds) as a vast collection of different worlds. Mathematicians have a big mystery called the L-space Conjecture. It suggests that these worlds can be sorted into two camps based on three different "tests":

1. The Flow Test: Does the world have a special kind of "wind" (a taut foliation) that flows smoothly everywhere without getting stuck?
2. The Order Test: Can you arrange the "symmetries" (the ways you can rotate or shift the world) into a neat, non-circular line (like a queue)? This is called being left-orderable.
3. The Topology Test: Is the world "simple" in a specific algebraic sense (an L-space)?

The conjecture says: If a world passes the Flow Test, it must also pass the Order Test.

Bojun Zhao's paper is about proving this connection. He wants to show that if you have a specific type of "wind" (a taut foliation) in a 3D world, you can automatically organize the symmetries of that world into a neat line.

The Problem: The "Traffic Jam" in the Leaf Space

To understand the proof, we need to zoom in on the "wind" (the foliation). Imagine the 3D world is made of infinite sheets of paper stacked on top of each other.

• If you look at this from a distance, the sheets look like a smooth stack.
• But if you zoom in to the "universal cover" (a giant, infinite version of the world), the sheets might get weird. Sometimes, two sheets might get so close they touch at a point but never cross, or they might branch out like a tree.

Mathematicians call this mess a Cataclysm. Think of a cataclysm like a traffic jam where cars (leaves) are bumper-to-bumper but can't pass each other. In a normal, well-behaved stack, you can easily say "Car A is above Car B." But in a traffic jam (cataclysm), it's hard to tell who is who.

The Challenge: To prove the symmetries can be ordered, you need to be able to say "Symmetry A comes before Symmetry B." But if the underlying structure (the leaves) is jammed up in a cataclysm, you can't easily define an order.

The Solution: "Orderable Cataclysms"

Zhao introduces a special condition: Orderable Cataclysm.

Imagine that even though the traffic is jammed, there is a traffic cop (a specific rule) standing at the jam. This cop has a strict rule: "In this specific jam, the red cars are always considered 'before' the blue cars, and the blue cars are 'before' the green cars."

If every traffic jam in the entire infinite world has such a traffic cop who follows a consistent rule, then the whole system is Orderable.

The Main Theorem:
Zhao proves: If your 3D world has a "wind" (foliation) where every traffic jam (cataclysm) has a consistent traffic cop (orderable cataclysm), then the symmetries of that world can be arranged in a perfect line.

The Analogy: The Infinite Library

Let's try a different analogy. Imagine an infinite library where books are arranged on shelves.

• The Leaves: The books.
• The Cataclysm: A section where the shelves are broken, and books are piled up in a way that makes it hard to tell which is "left" and which is "right."
• The Orderable Cataclysm: Even though the shelves are broken, someone has written a label on every pile saying, "This pile is sorted by height, then by color."

Zhao says: "If every broken pile in this infinite library has a clear sorting rule, then I can take the entire library and arrange every single book in one giant, perfect line from start to finish."

Why This Matters: The "Anosov Flow" Connection

The paper applies this to a famous type of wind called an Anosov flow (think of a chaotic but predictable weather pattern, like a hurricane that never stops spinning).

• Previous Proof: Before this, mathematicians proved that Anosov flows lead to ordered symmetries, but they used a very heavy, complicated tool called the "Universal Circle." It was like using a nuclear-powered calculator to solve a simple math problem. It worked, but it was hard to understand why.
• Zhao's Proof: Zhao says, "We don't need the nuclear calculator." He shows that you can just look at the "ends" of the leaf space (the tips of the branches of the tree) and the traffic jams. If the traffic jams are orderly, the ends of the branches naturally fall into a line. This is a much simpler, more "elementary" explanation.

The Second Part: Dehn Filling (The "Surgery" on the World)

The second half of the paper talks about Dehn Filling. Imagine you have a 3D world with some "knots" or "holes" (singular orbits). You can perform surgery on these holes by gluing them shut in different ways.

Zhao proves that if you start with a world that has a "good" wind (pseudo-Anosov flow) and you perform surgery on the knots using a specific "tight" angle (where the cut intersects the knot exactly once), the new world you create will still have an orderly wind.

The Takeaway: You can take a messy, chaotic world, cut it open, and glue it back together in a specific way, and the "Order" property will survive the surgery. This means there are infinitely many new worlds that satisfy the L-space conjecture.

Summary in One Sentence

Bojun Zhao shows that if a 3D shape has a "wind" where even the messy, jammed-up parts follow a consistent sorting rule, then the entire shape's symmetries can be lined up in a perfect, non-circular order, proving a major piece of the L-space puzzle without needing overly complex machinery.

Technical Summary

1. Problem Statement and Context

The paper addresses a central problem in low-dimensional topology and geometric group theory: the relationship between the existence of specific geometric structures on 3-manifolds and the algebraic property of left-orderability of their fundamental groups.

This work is situated within the framework of the L-space conjecture (proposed by Boyer-Gordon-Watson and Juhász), which posits the equivalence of three properties for a connected, closed, orientable, irreducible 3-manifold $M$:

1. $M$ is not an L-space (a Heegaard Floer homology invariant).
2. The fundamental group $\pi_1(M)$ is left-orderable.
3. $M$ admits a co-orientable taut foliation.

While the implication (3) $\Rightarrow$ (1) is known, the implication (3) $\Rightarrow$ (2) (that a co-orientable taut foliation implies left-orderability) remains a major open conjecture. Previous proofs for specific cases often relied on Thurston's universal circle action, a sophisticated tool involving hyperbolic geometry and dynamical systems.

The Core Question: Can one prove that $\pi_1(M)$ is left-orderable given a co-orientable taut foliation using more elementary methods, specifically avoiding the universal circle action? Furthermore, does this hold for specific classes of foliations arising from Anosov and pseudo-Anosov flows?

2. Methodology

Zhao develops a purely combinatorial and topological approach based on the leaf space of the lifted foliation in the universal cover.

• Leaf Space Analysis: Let $\mathcal{F}$ be a taut foliation on $M$, and $\tilde{\mathcal{F}}$ its pull-back to the universal cover $\tilde{M}$. The leaf space $L(\mathcal{F})$ is a simply connected, possibly non-Hausdorff 1-manifold. The fundamental group $\pi_1(M)$ acts on this leaf space.
• Cataclysms: The author utilizes the concept of a cataclysm (a maximal set of non-separated points in the leaf space). These represent "branching" points where the leaf space fails to be Hausdorff.
• Orderable Cataclysm: A key definition introduced is orderable cataclysm. A foliation has orderable cataclysm if, for every cataclysm $\mu$, there exists a linear order on the points of $\mu$ that is invariant under the stabilizer of $\mu$ in $\pi_1(M)$.
• Construction of the Order:
  1. The author constructs "broken paths" ($\alpha(u, v)$) between points in the leaf space, which traverse through cataclysms.
  2. These paths are analyzed for "cusps" (transitions between cataclysms).
  3. A numerical invariant $n(x_1, x_2)$ is defined for pairs of ends (positive ends) of the leaf space, counting the difference between positive and negative cusps along the broken path connecting them.
  4. Using the properties of these broken paths and the orderable cataclysm condition, the author proves that $n(x_1, x_2)$ defines a linear order on the set of positive ends of the leaf space.
  5. This linear order is shown to be preserved by the action of $\pi_1(M)$.
• Group Theory: By establishing a non-trivial action of $\pi_1(M)$ on a linearly ordered set (the ends of the leaf space), the author invokes standard results (e.g., from Boyer-Rolfsen-Wiest) to conclude that $\pi_1(M)$ is left-orderable.

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

If $M$ admits a co-orientable taut foliation $\mathcal{F}$ with orderable cataclysm, then $\pi_1(M)$ is left-orderable.

• Significance: This provides an elementary proof for left-orderability that relies solely on the structure of the leaf space and does not require Thurston's universal circle action or the assumption that the manifold is atoroidal.

Application to Anosov Flows (Corollary 1.2)

• Fenley previously proved that stable and unstable foliations of Anosov flows always have orderable cataclysms.
• Result: If $M$ admits an Anosov flow with a co-orientable stable foliation, then $\pi_1(M)$ is left-orderable.
• Refinement: Even if the stable foliation is not co-orientable, $\pi_1(M)$ has a subgroup of index 2 that is left-orderable.
• Novelty: This recovers known results about Anosov flows but via a direct, elementary argument involving the leaf space ends, bypassing the complex machinery of the universal circle.

Application to Pseudo-Anosov Flows and Dehn Fillings (Proposition 1.5)

• The paper considers pseudo-Anosov flows with co-orientable stable foliations.
• Result: For a manifold $M$ with such a flow, performing Dehn filling along the singular orbits with slopes that are "transverse" (intersection number 1) to the preferred framing yields a new manifold $N$.
• Conclusion: These filled manifolds $N$ admit co-orientable taut foliations with orderable cataclysms, implying their fundamental groups are left-orderable.
• Scope: This applies to infinitely many Dehn fillings, extending previous results (e.g., by Zung) which were restricted to specific slope signs.

4. Technical Details of the Proof

The proof hinges on the construction of a linear order on the set of ends of the leaf space, denoted $End^+(L)$.

1. Broken Curves: The author defines a unique "broken curve" $\alpha(x_1, x_2)$ between two ends. This curve consists of segments in the leaf space and "jumps" across cataclysms.
2. Cusp Counting: The orientation of the segments changes at every cusp. The author defines a count $n(x_1, x_2)$ based on the net number of positive vs. negative cusps.
3. Triangle Inequality: A crucial lemma (Lemma 3.11) establishes that for three ends $x_1, x_2, x_3$, the value $n(x_1, x_3)$ is either $n(x_1, x_2) + n(x_2, x_3) \pm 1$. This "almost additive" property, combined with the fact that $n(x_1, x_2) \neq 0$, allows the definition of a strict linear order: $x_1 < x_2$ if and only if $n(x_1, x_2) < 0$.
4. Equivariance: The order is shown to be invariant under the $\pi_1(M)$ action because the order on cataclysms is invariant by assumption, and the cusp counting respects the group action.

5. Significance

• Elementary Proof: The paper provides a significant conceptual simplification for proving left-orderability in the context of Anosov flows. It demonstrates that the deep dynamical information encoded in the universal circle is not strictly necessary for this specific algebraic conclusion; the combinatorial structure of the leaf space suffices.
• Generalization: It extends the scope of known left-orderable manifolds to a vast class of Dehn fillings of pseudo-Anosov flows, which are central to the study of 3-manifold topology.
• L-space Conjecture: While not proving the full conjecture, it strengthens the evidence for the (3) $\Rightarrow$ (2) implication by identifying a broad class of taut foliations (those with orderable cataclysm) that satisfy the condition.
• Methodological Innovation: The technique of using "broken paths" and cusp counting on non-Hausdorff leaf spaces offers a new toolkit for analyzing the interaction between group actions and foliation topology.

In summary, Bojun Zhao successfully bridges the gap between the geometric structure of taut foliations and the algebraic property of left-orderability using a novel, self-contained argument centered on the orderability of cataclysms in the leaf space.