Is there a good method to estimate the diameter of a closed hyperbolic 3-manifold?


I am particularly interested in know the diameter of the Weeks manifold.



Diameter of hyperbolic 3-manifolds
On page 356 of his paper *[The ortho-length spectrum for hyperbolic 3-manifolds](http://qjmath.oxfordjournals.org/content/47/3/349.extract)*, Meyerhoff shows that if $M$ is a closed hyperbolic $3$-manifold of volume $V$ then its diameter $\mathrm{diam}(M)$ satisfies $$\mathrm{diam}(M) < \frac{V}{\pi\sinh^2(\ell/4)},$$
where $\ell$ is the real length of the shortest geodesic of $M$. 


In their paper *[Symmetries, Isometries and Length Spectra of Closed Hyperbolic Three-Manifolds](https://projecteuclid.org/euclid.em/1048515809)*, Hodgson and Weeks compute the initial part of the length spectra of the ten smallest closed hyperbolic $3$-manifolds. If I am reading their Table 1 correctly, for the Weeks manifold you should have $\ell=0.58460369$. The volume of the Weeks manifold is $V=0.94270736$, hence we conclude from Meyerhoff's inequality that $$\mathrm{diam}(M)<13.9487057150346
.$$


