# The moduli space of finite volume hyperbolic 3-manifolds?

By finite volume hyperbolic 3-manifold, I do mean $$M=\mathbb{H}^{3}/\Gamma$$ where $$\Gamma$$ is a torsion-free Kleinian group such that the hyperbolic volume $$Vol(M)<\infty$$.

I will call $$\mathcal{M}:=\{\textrm{finite volume hyperbolic 3-manifolds}\}/_{\cong_ {isometry}}$$

The moludi space of finite volume hyperbolic 3-manifolds. As far as I understand, the hyperbolic volume function $$Vol: \mathcal{M} \rightarrow \mathbb{R}^{+}$$ has some nice properties: the image of this function is closed and well-ordered subset of $$\mathbb{R}^{+}$$ and the preimage $$Vol(x)^{-1}$$ is finite for any $$x\in \mathbb{R}^{+}$$

I have no knowledge in this field, but out of curiosity I wanted to know if there is some research done in understanding $$\mathcal{M}$$, is there a natural topological structure on it? is $$\mathcal{M}$$ an manifold or an orbifold ? Can we endow $$\mathcal{M}$$ with some algebraic or geometrical structure ?

• The rigidity of hyperbolic 3-manifolds means that the only natural topology on $\mathcal M$ is zero-dimensional. With the discrete topology, it is a manifold in the trivial sense of a set with the discrete topology. There may also be a natural topology which is still totally disconnected but not discrete. Aug 2, 2021 at 23:04
• Indeed, a nontrivial topology is given by en.wikipedia.org/wiki/Geometric_topology_(object) It is still totally disconnected, and thus can't really be said to be a geometric (or algebraic) structure Aug 3, 2021 at 0:22
• @WillSawin the link looks very interesting! Thanks
– GSM
Aug 3, 2021 at 0:46

By Mostow rigidity theorem two hyperbolic manifolds are isometric if they have the same fundamental group, so there are really no deformations and, thus, manifold structure.

• But how about some algebraic or geometric structure on $\mathcal{M}$ ?
– GSM
Aug 2, 2021 at 23:17

One approach is to use the geometric topology: that is, the topology of Gromov-Hausdorff convergence. This is closely related to Thurston's hyperbolic Dehn surgery theorem.

There are further approaches to producing a "space" of three-manifolds. In this video Thurston alludes to cone-manifold deformations and commensurability relations. I vaguely remember that he also had a kind of "solenoid" space that contained all hyperbolic three manifolds (finite and infinite volume) as leaves. I can't find references to this on-line - perhaps another reader will help us.

• The "solenoid space" you refer to is probably related to the space of pointed hyperbolic manifolds (with pointed Hausdorff topology) where each single hyperbolic manifold gives a "leaf" in there via the map $M \ni x \mapsto (M, x)$. Aug 5, 2021 at 14:45
• cone-manifold deformations give a geometric interpretation for the deformation space of a cusped manifold, which is a manifold "filling in" the space between the manifolds obtained by Dehn filling the cusps (which is a 0-dimensional subset). A good reference on this deformation space (besides Thurston's original notes) is the paper of Neumann and Zagier (zbmath.org/?q=an%3A0589.57015) Aug 5, 2021 at 14:50
• @JeanRaimbault - Is there are reference for this? And in particular some discussion of Thurston's? I definitely remember this coming up in talks (in the 90's and 00's) but I could not find anything online. Aug 24, 2021 at 17:35
• regarding the space of 3--manifolds you can find some explanations, together with examples of solenoidal subspaces (minimal laminations with transverse space a Cantor) in this paper arxiv.org/pdf/1612.09510.pdf ; I'm not aware of earlier references on this specific topic though this was certainly well-known. Sep 2, 2021 at 15:05
• Thank you for the reference. Sep 2, 2021 at 15:13