Connectivity in the space of framed hyperbolic 3-manifolds

We prove that the space $\mathcal{H}\infty$ of framed infinite volume hyperbolic $3$-manifolds is connected but not path connected. Two proofs of connectivity of this space, which is equipped with the geometric topology, are given, each utilizing the density theorem for Kleinian groups. In particular, we construct a hyperbolic $3$-manifold whose set of framings is dense in $\mathcal{H}\infty$. Examples of paths in $\mathcal{H}\infty$ are discussed, including paths of geometrically finite manifolds limiting to certain infinite type geometric limits of quasi-Fuchsian manifolds. The discussion of paths culminates in describing an infinite family of non-tame hyperbolic $3$-manifolds, each of whose set of framings is a path component of $\mathcal{H}\infty$, establishing that $\mathcal{H}_\infty$ is not path connected.

💡 Research Summary

The paper studies the global topology of the space 𝓗 of framed hyperbolic 3‑manifolds, focusing on the subspace 𝓗_∞ consisting of infinite‑volume manifolds. By identifying 𝓗 with the Chabauty space 𝔇 of torsion‑free Kleinian groups (via the bijection Φ that sends a group Γ to the framed manifold (H³/Γ, π_Γ(O))) the authors equip 𝓗 with the geometric topology, which is generated by (ε,R)‑closeness of framed manifolds.

The main results are:

Theorem A – The connected components of 𝓗 are exactly the leaves ℱ(M) for each finite‑volume hyperbolic 3‑manifold M, together with the whole infinite‑volume subspace 𝓗_∞. The finiteness case follows from Mostow–Prasad rigidity; the infinite‑volume case requires more work.

Connectivity of 𝓗_∞ – Two independent proofs are given. First, Lemma 3.3 shows that every geometrically finite infinite‑volume framed manifold lies in the same path component as the base point (ℍ³, O). The density theorem of Namazi–Souto and Ohshika guarantees that this path component is dense in 𝓗_∞, so its closure equals 𝓗_∞, establishing connectivity.

Second, Theorem B constructs a specific infinite‑volume manifold M by modifying the circle‑packing construction of Fuchs, Purcell, and Stewart. Proposition 3.8 proves that any infinite‑volume framed manifold is a geometric limit of framed manifolds whose convex‑core boundaries consist of disjoint totally geodesic thrice‑punctured spheres. Gluing together such building blocks yields a manifold M whose leaf ℱ(M) is dense in 𝓗_∞, providing an alternative proof of connectivity.

Path analysis – The authors develop a machinery for tracking individual group elements along a path in the Chabauty space. Proposition 4.2 defines, for a path Γ:I→𝔇, a family of maps J_{s,t}:Γ_s→PSL₂ℂ with the properties that (i) J_{s,s} is the identity, (ii) J_{s,t}(ψ) lies in Γ_t or is the point at infinity, (iii) the trajectory t↦J_{s,t}(ψ) is continuous, and (iv) new elements can only appear by “coming from infinity.” This formalism makes precise the phenomenon that, along a path, elements may disappear to infinity or be introduced from infinity, a behavior illustrated using Klein–Maskit combination theorems (Example B.3).

Theorem C constructs a family of paths G: