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The geometrization theorem tells us:

Theorem (Thurston) The mapping torus $M_\phi$ of a pseudo-Anosov diffeomorphism $\phi: S_g \rightarrow S_g$ from a genus $g$ surface to itself admits a complete hyperbolic metric of finite volume.

Otal has a complete presentation of this result starting from standard graduate material.

My question is: How much can one say about the geometry of the $M_\phi$ as a function of $\phi$ and $g$? Are there upper bounds on volume? Systole? Are there simpler bi-Lipschitz models of $M_g$? Are there any easily computable geometric quanities?

For example -- One has that the Heegaard genus of $M_g$ is $g$. This gives a lower bound on volume in terms of $g$. Is there an upper bound?

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To address your last point, for a mapping torus of $S_g$, there is an upper bound on the Heegaard genus of $2g+1$, which is sharp (in fact, $M_{\phi^n}$ will have rank $2g+1$ for $n$ large), but is not always an equality. However, there is no upper bound on the volume of a fiber bundle in terms of Heegaard genus. For example, Jesse Johnson has a description of all fiber bundles of Heegaard genus 2, and it's not hard to see that these examples may have arbitrarily large volume by a Dehn filling argument.

For a fairly explicit and detailed description of the relation between the geometry and the monodromy for punctured torus bundles (in particular, its continued fraction expansion), see the work of Gueritaud. One has that the volume is coarsely related to the continued fraction expansion length (or LR-decomposition, which is the analogue of Brock's pants distance result in this case), with explicit sharp constants. This is preceded by the work of Minsky (who gets coarse estimates on the holonomies of pivot curves) and in fact Jorgensen who originally discovered hyperbolic structures on these manifolds. See also this paper for estimates on the cusp area in terms of the LR-length and other geometric information.

Unfortunately in the higher genus case, there is not such a good comparison of the topological and geometric pictures. In particular, there are not many results which are uniform in the genus (although Brock's upper bound on volume in terms of pants distance has a uniform constant, but the lower bound cannot). For fixed genus, Minsky's model in principle gives a complete coarse comparison of the geometry with the topological description in terms of the action of the monodromy on the curve complex. However, the constants are not explicit, as they are obtained by geometric limit arguments. Also, there is not a strict analogue of continued fraction length; instead, one works with ``tight geodesics" in the curve complex, which are difficult to compute with. All this machinery was developed to resolve the ending lamination conjecture, but of course applies to the fibered case.

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There is a considerable literature on the subject, where (among) the names to look up are Jeff Brock, Hossain Namazi, Juan Souto, David Futer, Saul Schleimer. Some of the work uses the machinery of the curve complex, which you can also google.

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  • $\begingroup$ Thanks for the great references. I've looked in to the work by Namazi-Souto and will dig deeper. The others I only know tangentially. $\endgroup$ – pgadey Nov 6 '13 at 15:28
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At least a coarse answer to the first question: http://www.math.brown.edu/~brock/home/text/papers/3ms1/www/3ms1.pdf shows that the volume of the mapping torus can be bounded above and below and above in terms of the monodromie's translation distance (with respect to the Weil-Petersson metric on Teichmüller space).

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