teichmuller geodesics and hyperbolic mapping torus

Question

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 $\phi$.

Answer 1

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.

Answer 2

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.

Answer 3

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 $\sigma$.