Connectivity in the space of framed hyperbolic 3-manifolds

This paper establishes that the space of framed infinite volume hyperbolic 3-manifolds is connected but not path connected by proving its connectivity via the density theorem for Kleinian groups and demonstrating the existence of non-tame manifolds whose framing sets form distinct path components.

The Gist

Imagine you are an explorer trying to map a vast, invisible universe made of shapes. This universe isn't made of stars and planets, but of hyperbolic 3-manifolds.

To put it simply: think of a hyperbolic 3-manifold as a piece of space that curves away from itself in every direction, like the inside of a Pringles chip, but extended into three dimensions. These shapes can be finite (like a donut) or infinite (like an endless, twisting tunnel).

Now, imagine we attach a tiny, 3D compass to every single point in these shapes. This compass tells us exactly which way is "up," "forward," and "right." In math, this is called a framed manifold.

The paper by Matthew Zevenbergen is about exploring the "Space of All Possible Shapes." Let's call this the Grand Gallery.

The Grand Gallery: Connected but Not Walkable

The author asks a big question: Is the Grand Gallery one big, connected room where you can walk from any shape to any other shape without jumping?

In math, there are two ways to be "connected":

1. Connected (The "Blob" Test): Can you get from Point A to Point B if you are allowed to teleport or jump over tiny gaps? (Like a sponge: it's all one piece, even if it has holes).
2. Path Connected (The "Walk" Test): Can you draw a single, continuous line (a path) from Point A to Point B without ever lifting your pen?

The Big Discovery:
Zevenbergen proves that the Grand Gallery of infinite shapes is Connected (it's all one piece), but it is NOT Path Connected (you cannot walk a continuous path between certain shapes).

It's like a sponge that is glued together so tightly you can't pull it apart, but the holes are so deep and narrow that you can't swim through them. You can get close to any shape, but you can't always travel to it.

The Two Proofs: How He Figured It Out

1. The "Dense Crowd" Proof (Connectivity)
Imagine a party where most people are standing in a huge, dense crowd in the center of the room. Even if there are a few people standing alone in the corners, if the crowd is dense enough, the whole room feels "connected."

• Zevenbergen uses a famous math theorem (the Density Theorem) to show that there is a specific type of "simple" shape that can be tweaked to look almost exactly like any complex, infinite shape.
• Because these simple shapes are everywhere (dense), and they are all connected to each other, the whole room of infinite shapes is connected.

2. The "One-Man Band" Proof (The Dense Leaf)
He then constructs a specific, incredibly complex shape (let's call it The Master Shape).

• He proves that if you take this Master Shape and rotate its compass (change its "frame"), you can get infinitely close to any other infinite shape in the universe.
• It's like having a single, magical origami paper that, if you fold it just right, can mimic the shape of a bird, a boat, or a castle. Because this one paper can mimic everything, the space is connected.

The "Walk" Problem: Why You Can't Get There From Here

This is the most surprising part. Even though the room is connected, you can't walk between certain shapes.

The "Symmetric Glue" Shapes:
Zevenbergen invents a family of shapes made by gluing together copies of a small, rigid block (like a Lego brick) in a perfectly symmetrical, infinite pattern. Let's call these The Infinite Lego Castles.

• The Rigidity: These castles are so perfectly symmetrical and rigid that they have only one possible way to exist in hyperbolic space. They are like a perfect diamond; you can't squeeze or stretch them without breaking them.
• The Trap: If you try to walk (create a path) from one of these castles to a slightly different shape, the math says you can't. To change the shape, you would have to "break" the symmetry or stretch the space in a way that is impossible for these specific rigid blocks.
• The Result: Each of these Infinite Lego Castles is stuck in its own little "island" in the Grand Gallery. You can't walk off the island to get to another one.

The Analogy of the "Infinite Train"

Imagine a train track that stretches forever.

• Connected: The track is one continuous piece of metal.
• Not Path Connected: Imagine that at certain stations, the tracks are separated by a gap that is too wide to jump, but the metal is still technically "connected" because the gaps are filled with invisible glue (mathematical density).
• The Specific Stations: The "Infinite Lego Castles" are like stations where the tracks are locked. You can stand on the platform, but the train (your path) cannot leave the station to go anywhere else.

Why Does This Matter?

In the world of 2D shapes (like flat surfaces), everything is connected and you can walk anywhere. But in 3D hyperbolic space, the rules change. The universe of shapes is much more complex.

• For Mathematicians: This settles a long-standing debate about the structure of these spaces. It shows that "being close" (connected) is very different from "being able to travel" (path connected).
• For the Rest of Us: It's a reminder that in complex systems, things can be intimately related and part of the same whole, yet completely isolated from one another in terms of movement. It's like a social network where everyone knows everyone else (connected), but some people are so locked in their own routines that they can't actually meet up with anyone else (not path connected).

In a nutshell: The universe of infinite 3D shapes is one big, tangled web. You can get close to anything, but for some very special, rigid shapes, you are stuck in your own little corner, unable to walk to the next one.

Technical Summary

1. Problem Statement

The paper investigates the global topological properties of the space $\mathcal{H}$, which consists of isometry classes of complete, oriented, connected, framed hyperbolic 3-manifolds $(M, f)$. The space is equipped with the geometric topology (equivalent to the Chabauty topology on the space of Kleinian groups).

The central questions addressed are:

1. Connectivity: What are the connected components of $\mathcal{H}$?
2. Path Connectivity: Is $\mathcal{H}$ path-connected? Specifically, is the subspace $\mathcal{H}_\infty$ (manifolds of infinite volume) path-connected?

While the finite volume case is well-understood via Mostow rigidity, the infinite volume case presents significant challenges due to the flexibility of geometric limits and the behavior of paths in the space of Kleinian groups.

2. Methodology

The author employs a combination of classical deformation theory, geometric group theory, and the recently resolved Density Theorem for Kleinian groups.

• Geometric Topology & Chabauty Convergence: The analysis relies on the definition of convergence for sequences of Kleinian groups $\{\Gamma_n\} \to \Gamma$. This requires two conditions:
  1. Any accumulation point of elements in $\Gamma_n$ must lie in $\Gamma$.
  2. Every element in $\Gamma$ must be the limit of a sequence in $\Gamma_n$.
• Density Theorem: The author utilizes the strong version of the Density Theorem (Namazi-Souto, Ohshika), which states that every Kleinian group is the geometric limit of a sequence of geometrically finite Kleinian groups.
• Path Tracking Machinery: A novel technical tool is developed to analyze paths in the space of Kleinian groups. For a path $\Gamma: [0,1] \to \mathcal{D}$, the author constructs maps $J_{s,t}: \Gamma_s \to \text{PSL}_2(\mathbb{C}) \cup \{\infty\}$. These maps track how group elements evolve along the path. If an element "diverges to infinity" (maps to $\infty$), it indicates a topological change in the manifold structure (e.g., a cusp pinching or a new handle appearing).
• Rigidity and Symmetry: The proofs for non-path-connectedness rely on constructing highly symmetric, non-tame manifolds (infinite type $(G, N)$-glued manifolds) and using rigidity theorems (Thurston's "AH is compact" for acylindrical manifolds) to show that paths cannot leave the conjugacy class of the starting group.

3. Key Contributions and Results

A. Connected Components of $\mathcal{H}$ (Theorem A)

The paper completely characterizes the connected components of the space $\mathcal{H}$:

1. Finite Volume: For each finite volume hyperbolic 3-manifold $M$, the set of its framings $\ell(M)$ forms a distinct connected component. This follows from Mostow-Prasad rigidity.
2. Infinite Volume: The set of all infinite volume framed hyperbolic 3-manifolds, denoted $\mathcal{H}_\infty$, constitutes a single connected component.

Proof Strategy for Connectivity of $\mathcal{H}_\infty$:
The author proves $\mathcal{H}_\infty$ is connected by showing it is the closure of a path-connected subset.

• Step 1: All geometrically finite infinite volume manifolds lie in the same path component as the standard hyperbolic space $(\mathbb{H}^3, O)$.
• Step 2: By the Density Theorem, geometrically finite groups are dense in the space of all Kleinian groups.
• Step 3: Since the closure of a connected set is connected, $\mathcal{H}_\infty$ is connected.

B. Existence of a Dense Leaf (Theorem B)

The paper constructs a specific hyperbolic 3-manifold $M$ such that its leaf $\ell(M)$ (the set of all framings of $M$) is dense in $\mathcal{H}_\infty$.

• Construction: This involves modifying a circle packing construction (Fuchs, Purcell, Stewart) to create a manifold whose convex core boundary is a disjoint union of totally geodesic thrice-punctured spheres.
• Significance: This provides a second proof of the connectivity of $\mathcal{H}_\infty$ and demonstrates that a single manifold type can approximate any infinite volume framed manifold arbitrarily well in the geometric topology.

C. Non-Path-Connectedness of $\mathcal{H}_\infty$ (Theorem D & Corollary E)

The most significant result is that while $\mathcal{H}_\infty$ is connected, it is not path-connected.

• Counter-Example Construction: The author defines an infinite family of symmetric infinite type $(G, N)$-glued hyperbolic 3-manifolds. These are constructed by gluing copies of a compact, acylindrical 3-manifold $N$ along their boundaries according to a highly symmetric infinite graph $G$.
• Rigidity Argument:
  1. These manifolds possess a unique hyperbolic structure due to their symmetry and acylindricity (Proposition 4.8).
  2. The author proves that if a path starts at such a manifold $M$, the tracking maps $J_{0,t}$ must remain injective homomorphisms for all $t$.
  3. If the path were to leave the conjugacy class of $\Gamma_0$, it would require elements to diverge to infinity. However, the acylindricity and bounded injectivity radius of these specific manifolds prevent this divergence without violating Thurston's compactness theorems or discreteness conditions.
• Conclusion: The leaf $\ell(M)$ for such a manifold is a path component of $\mathcal{H}_\infty$. Since there are infinitely many such non-isometric manifolds, $\mathcal{H}_\infty$ contains infinitely many path components.

D. Path Examples and Limit Behavior

The paper discusses various types of paths in $\mathcal{H}$:

• Moving Frames: Paths where the manifold is fixed, but the frame moves (conjugation in $\text{PSL}_2(\mathbb{C})$).
• Geometric Limits: Paths where geometrically finite manifolds limit to non-tame infinite volume manifolds (e.g., limits of quasi-Fuchsian manifolds).
• Cusp Rank Changes: Paths where a rank-1 cusp degenerates or expands into a rank-2 cusp.
• Non-Tame Limits: Construction of paths converging to manifolds homeomorphic to $(S \times \mathbb{R}) \setminus C$, where $C$ is a collection of curves, demonstrating that paths can connect geometrically finite manifolds to non-tame ones.

4. Significance

• Distinction between Connectivity and Path-Connectivity: The paper provides a striking example in geometric topology where a space is connected but not path-connected. This contrasts with the 2-dimensional case (Chabauty space of $\text{PSL}_2(\mathbb{R})$), which is path-connected (Warakkagun).
• Understanding Infinite Volume Manifolds: It clarifies the global structure of the deformation space of infinite volume hyperbolic 3-manifolds, showing that while they are topologically connected via limits, they are "separated" by path constraints due to rigidity.
• New Tools for Path Analysis: The development of the $J_{s,t}$ tracking maps provides a robust framework for studying continuous paths in the Chabauty space, distinguishing them from sequential convergence.
• Rigidity in Infinite Settings: The results highlight how strong rigidity properties (like acylindricity and symmetry) can persist even in infinite volume, non-tame settings, preventing certain deformations that are possible in more flexible geometric contexts.

In summary, Zevenbergen resolves the connectivity question for framed hyperbolic 3-manifolds, proving that the infinite volume sector is a single connected component but a complex, non-path-connected space containing infinitely many distinct path components defined by highly symmetric, non-tame manifolds.