Connectivity in the space of framed hyperbolic 3-manifolds

Abstract

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.

Figures (4)

• Figure 1: This depicts a portion of a circle packing $P$ with its associated triangulation. The shaded regions are curvilinear triangles in $\partial_{\infty}(\mathbb{H}^3/\Gamma)$. The dotted circles are the dual circles in $P^*$, including $c^{(v,u,w)}$.
• Figure 2: Some possible structures for the $\partial M_i$'s are depicted, along with the first few gluings in the construction of $M'$.
• Figure 3: This depicts the construction of $(\overline{M}',P')$ from $(\overline{M},P)$ in Example B.2, depicting $S$ as a genus-$2$ surface and the paring as shaded.
• Figure 4: This depicts a portion of the construction of a $(G,N)$-glued hyperbolic $3$-manifold, with $G$ overlaying the copies of $N$. The edges of $G$ labeled with a $1$ are depicted as dashed and those labeled with a $2$ are depicted solid.

Theorems & Definitions (51)

• Theorem 1
• Theorem 2
• Proposition
• Theorem 3
• Theorem 4
• Corollary 5
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Lemma 2.5