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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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1 Answer 1

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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 $$\mathrm{diam}(M)<13.9487057150346 .$$

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