# 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"  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"  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?

• 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