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.
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2$\begingroup$ Theorem 9.1(2) of this paper gives a crude upper bound in terms of the presentation length (the constant is not explicit, but it wouldn't be hard to compute an explicit constant which is likely far from sharp). ams.org/journals/jams/2012-25-01/S0894-0347-2011-00711-X/… $\endgroup$– Ian AgolCommented Sep 12, 2015 at 13:27
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On page 356 of his paper The ortho-length spectrum for hyperbolic 3-manifolds, 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, 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 $$\mathrm{diam}(M)<13.9487057150346 .$$