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Let $M_{\phi}$ be a hyperbolic mapping torus coming from a pseudo-Anosov map $\phi$ in a surface $S$. Is there any way to estimate the length of the geodesic representing a given curve in the surface in terms of the map $\phi$? That is, knowing something like the stable and unstable foliations for the map or something equivalent, can you estimate the length of a given curve? Any references for something like this are really appreciated.

For example, if you take a mapping torus $M_{\phi}$, drill one simple nontrivial curve $\alpha$ in the surface and re-glue by $\sigma^n$, a large Dehn twist about $\alpha$, you are going to get a hyperbolic mapping torus $M_{\phi\sigma^{n}}$. In this manifold $\alpha$ is going to be very short.

Another example, if you take a map $\psi = \phi\sigma^n$ where $\phi$ is pseudo-anosov in all of $S$ and $\sigma$ is a pseudo-Anosov just in a subsurface $X \subset S$, I think the curves in the complement of $X$ have to be very small for $n$ large, right?

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Regarding the last paragraph - Suppose $\phi$ is pA in all of $S$, and $\sigma$ is pA in $X$, a strict subsurface. The lengths of the curves of $\partial X$ go to zero in $M_{\phi \sigma^n}$, as $n$ goes to infinity. But for curves in the complement of $X$, yet not in the boundary, their length does not go to zero. – Sam Nead Mar 19 '12 at 11:34
(I edited your post to make the notation in the last paragraph match the previous paragraphs.) – Sam Nead Mar 19 '12 at 11:44
Thanks for that and for your answer. Do you know any example where the curves in the complement of $X$ don't have short length? – shurtados Mar 25 '12 at 19:23
Yes - this always happens if $S - X$ is more than a collection of annuli and pairs of pants. – Sam Nead Apr 26 '15 at 19:51
up vote 3 down vote accepted

In this paper, McMullen gets a coarse description of geodesics in a mapping torus of a punctured torus in terms of the Minsky model. I think one ought to get an estimate of their lengths from this.

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You should look at the papers of Yair Minsky. Perhaps the right place to start is "End invariants and the classification of hyperbolic 3-manifolds".

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