Connectivity in the space of framed hyperbolic 3-manifolds

TL;DR

This paper investigates the topological structure of the space of framed infinite volume hyperbolic 3-manifolds, proving it is connected but not path connected, and explores the properties of paths and framings within this space.

Contribution

It provides two proofs of the connectivity of the space, constructs a dense set of framings, and demonstrates the non-path-connectedness through examples of non-tame manifolds.

Findings

The space of framed infinite volume hyperbolic 3-manifolds is connected.

The space is not path connected, with explicit examples of non-tame manifolds.

A dense set of framings exists within the space.

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.