# Computing the Volume of Closed 3-Manifolds and the Geometrization Conjecture

My question is whether or not if I generalize Theorem 2(i) of "Contact Graphs of Unit Sphere Packings Revisited" [2012] by K. Bezdek and S. Reid (arXiv link) which states

The number of touching triplets (resp., quadruples) in an arbitrary packing of $n \geq 3$ (resp., $n \geq 4$) unit balls in $\mathbb{E}^3$ is at most $\frac{25}{3}n$ (resp., $\frac{11}{4}n$).

by replacing $\mathbb{E}^3$ by any of the other Thurston Geometries $\mathbb{S}^2 \times \mathbb{R}, \mathbb{H}^2 \times \mathbb{R}, \mathbb{S}^3, \mathbb{H}^3, \widetilde{SL_{2}(\mathbb{R})}$, Nilgeometry, or Solvgeometry, if I can use the Geometrization Conjecture to say something about the volume of closed 3-manifolds.

A succint statement of the Geometrization Conjecture for my purposes would be that for any closed 3-manifold $\mathcal{M}$ there exists a decomposition (I think it is called the JSJ-torus decomposition, denoted by $\otimes$) of $\mathcal{M}$ into prime 3-manifolds $\mathcal{N}_{i}$ (such a decomposition exists due to the Geometrization Conjecture recently proved by G. Perelman and neatly presented in "Completion of the Proof of the Geometrization Conjecture" [2008] by John Morgan and Gang Tian)

$$\mathcal{M} = \bigotimes_{i=1}^{n} \mathcal{N}_{i}$$

where each $\mathcal{N}_{i}$ admits one of the eight Thurston Geometries

$\mathbb{S}^2 \times \mathbb{R}, \mathbb{H}^2 \times \mathbb{R}, \mathbb{E}^3, \mathbb{S}^3, \mathbb{H}^3, \widetilde{SL_{2}(\mathbb{R})}$, Nilgeometry, or Solvgeometry.

and is of a finite volume.

My idea now is that the volume of each prime 3-manifold which $\mathcal{M}$ was decomposed into can have it's volume approximated by determining the maximum number of regular 3-simplices in a simplicial 3-complex $\mathcal{K}_{i}$ which can be embedded into the $i$-th prime 3-manifold in the decomposition. Then,

$$\text{vol}\left(\mathcal{M}\right) > \sum_{i=1}^{n} \text{vol}(\mathcal{K}_{i})$$

With a generalization of Theorem 2(i) to each of the Thurston Geometries, then I would be able to compute this bound by multiplying the maximum number of unit balls I can fit in the space by the volume of the unit ball and dividing by the optimal known packing density (note that a regular 3-simplex corresponds to a touching quadruple of spheres, which is why Theorem 2(i) would be useful). Does this general outline make sense? I don't know a lot about the decomposition of manifolds or the volume of manifolds, so any feedback on the idea or references would be appreciated. In particular, my question is:

If I do all of the work to get a version of Theorem 2(i) in each Thurston Geometry, can I use the Geometrization Conjecture for studying (in this example, I was thinking volume computation of 3-manifolds) some interesting properties of 3-manifolds?

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Samuel: Except for the hyperbolic geometry, volumes of all other geometric 3-manifolds are easily computable by using, say, Gauss-Bonnet formula in the case of the geometries fibered over the hyperbolic plane, etc. In the hyperbolic case, I do not see how your method would yield something interesting (comparing to Gromov-norm interpretation of volume or computation using an ideal triangulation). However, maybe I am missing something here. Also, take a look at the work of Gabai, Meyerhoff and Milley on volumes of hyperbolic 3-manifolds, it is also based on packing bounds. – Misha Feb 26 '13 at 20:10