# Pseudoanosov mapping torus and length of curves.

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?

• 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. Mar 19 '12 at 11:34
• (I edited your post to make the notation in the last paragraph match the previous paragraphs.) 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? Mar 25 '12 at 19:23
• Yes - this always happens if $S - X$ is more than a collection of annuli and pairs of pants. Apr 26 '15 at 19:51

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.

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".