# 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 Hausdorff 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.

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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
@shurtados: This is hard to answer without knowing what geometric information you are particularly looking for. But, what I was thinking of in the 1st paragraph of my answer is the standard description of Teichmuller geodesics, and the corresponding conformal structures and singular Euclidean metrics, which can be found in Farb Margalit, "A primer on mapping class groups". I am also fond of Abikoff's little book "The real analytic theory of Teichmuller space". Descriptions in terms of hyperbolic structures are implicit from the uniformization theorem, but are much harder to see directly. – Lee Mosher Mar 22 '12 at 16:59
Lee - as you suspected, no uniform estimate can exist. See my answer for a family of examples (which also cover the WP metric.) – Sam Nead Jun 6 '15 at 13:13

To expand on Lee's answer, recall that Teichmüller space can be divided up into a "thin part" (where some geodesic is short, or equivalently where some conformal annulus has large modulus), and its complement, the "thick part". I would guess that

1. The portions of $l$ and $\sigma$ that are in the thick part are a uniform distance from each other, and in particular if they don't enter the thin part then they are a bounded distance apart.

2. Inside the thin part, $l$ goes closer to the singularity (i.e., the length of the short geodesic gets shorter). (To the extent that $l$ is well-defined, anyway, I haven't thought that through.)

Jeff Brock proved that the hyperbolic volume of the mapping torus is approximated by the translation distance of $\phi$ in the Weil-Petersson metric on Teichmüller space. The Weil-Petersson and Teichmüller geodesics are quasi-isometric outside the thin part; inside the thin part, the Weil-Petersson metric puts the singularity at a finite distance. It's reasonable to guess that $l$ would track the geodesic corresponding to $\phi$ in the Weil-Petersson metric, but I'm not sure whether that's been proved or not.

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Dylan - $\ell$ need not track $\sigma$, even in the WP metric. See my answer for a sketch of this. – Sam Nead Apr 26 '15 at 20:11
The path $\ell$ is not well defined. That is, there is no best path of pleated surfaces around a mapping torus. Thus the answer to your question is "no": there is no upper bound on the Hausdorff distance between $\ell$ and $\sigma$, purely in terms of the topology of $S$.
To prove this, we resort to a standard example. Suppose that $S$ is the genus two surface. Suppose $X$ and $Y$ are disjoint subsurfaces of $S$, both homeomorphic to the once-holed torus. Let $\gamma = \partial X = \partial Y$.
Let $f$ be a pA map on $X$, let $g$ be a pA map on $Y$, and let $h$ be a pA map on $S$. Then for all sufficiently large $n$ the map $F_n = (f g)^n h$ is a pA map. Let $M_n$ be the mapping torus for $F_n$. Then the curve $\delta = \gamma \times \{1/2\} \subset M_n$ is very short in the hyperbolic metric in $M_n$. Consider the following two paths $\ell$ and $\ell'$ of pleated surfaces. The path $\ell$ first "moves through" $X$ and then moves through $Y$. The path $\ell'$ moves through $X$ and $Y$ in the opposite order. The paths $\ell$ and $\ell'$ are far apart in Teichmüller space, so at least one of them is far away from $\sigma$, as measured in Hausdorff distance. (The exact same construction works if we instead use the WP metric.)
Basically, the thin part of Teichmüller space contains large product-like regions. Teichmüller geodesics "know" how to go through such regions. Pleating paths, which are very far from being unique, do not have such knowledge. As a final remark - there are periodic Teichmüller geodesics that live completely in the thin part. Using such a $\sigma$ we can arrange an $\ell$ that is at no point close to $\sigma$.