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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 ?

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    $\begingroup$ 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. $\endgroup$
    – Will Sawin
    Commented Aug 2, 2021 at 23:04
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    $\begingroup$ 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 $\endgroup$
    – Will Sawin
    Commented Aug 3, 2021 at 0:22
  • $\begingroup$ @WillSawin the link looks very interesting! Thanks $\endgroup$
    – GSM
    Commented Aug 3, 2021 at 0:46

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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.

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

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    $\begingroup$ 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 $M \ni x \mapsto (M, x)$. $\endgroup$
    – Jean Raimbault
    Commented Aug 5, 2021 at 14:45
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    $\begingroup$ 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) $\endgroup$
    – Jean Raimbault
    Commented Aug 5, 2021 at 14:50
  • $\begingroup$ @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. $\endgroup$
    – Sam Nead
    Commented Aug 24, 2021 at 17:35
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    $\begingroup$ 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. $\endgroup$
    – Jean Raimbault
    Commented Sep 2, 2021 at 15:05
  • $\begingroup$ Thank you for the reference. $\endgroup$
    – Sam Nead
    Commented Sep 2, 2021 at 15:13

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