# teichmuller geodesics and hyperbolic mapping torus

Given a pseudoanosov map $\phi$ of a surface $S$, there is a geodesic $\sigma$ in Teichmuller space (with the teichmuller metric) that is an axis for $\phi$, In other words, $\phi$ acts as a translation along that geodesic. In the other hand, we can take the cyclic covering $\tilde M_\phi$ of the Mapping torus $M_\phi$, and get a path of pleated surfaces $S_t$ exiting both ends of $M_\phi$, that "coarsely" is going to give us a path $\pi$ in teichmuller space.

I would like to know what's the relation between these two curves $l$ and $\sigma$in teichmuller space, if it's possible to estimate the Haussdorff distance between them in terms of the genus of $S$.

I would also like to know if there is any way of describing how the points in $\sigma$ are going to look like, that is, to give some description of the surfaces in the axis of $\phi$.

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Points in the Teichmuller geodesic $\sigma$ are given quite concretely in terms of the pseudo-Anosov data, namely the stable and unstable measured foliations, by using the standard method one uses to describe any Teichmuller geodesic from a pair of measured foliations.
I'm guessing that there's not going to be any uniform estimate in terms of genus for Hausdorff distance between a pleated surface path and the Teichmuller geodesic. There exist examples in which $M_\phi$ has arbitrarily short geodesics, and a pleated surface passing near such a geodesic also will have arbitrarily short geodesics. I'm pretty sure in such a case that there would then be an arbitrarily large distance in Teichmuller space from the conformal structure on that pleated surface to the Teichmuller geodesic.
Thanks a lot for your answer, I really appreciate it, do you know where can I find a reference describing how all the surface in the geodesic between two points in teichmuller space $X$ and $Y$ are going to look geometrically, depending in how $X$ and $Y$ look like geometrically? –  shurtados Mar 19 '12 at 6:15