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Geodesics in $\mathbb H^2$ have the following properties:

  1. For every two points in the plane there exists a unique geodesic joining them.

  2. Every geodesic determines exactly two points on the boundary of $\mathbb H^2$.

  3. Conversely, every pair of points on $\partial \mathbb H^2$ determine a unique geodesic

  4. Any two different geodesics that travels at bounded distance are asymptotic.

How many of the above properties are true for Teichmuller geodesics on the Teichmuller space of a closed surface with Thurston's boundary? Could you also provide references, please?

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    $\begingroup$ You should look at math.uchicago.edu/~masur/chapter3.pdf especially at Theorem 3.8 $\endgroup$
    – ThiKu
    Commented Nov 27, 2014 at 7:51
  • $\begingroup$ I don't understand point 4. What does it mean to "travel at bounded distance"? In $\mathbb H^2$, if you can travel at bounded distance in one direction, then you cannot in the other direction (unless the geodesics coincide). So does "travelling" refer only to future time, not past time? $\endgroup$
    – Matt
    Commented May 7, 2020 at 13:43

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I have answers to question 1,2 and 3.

1) True. This is essentially a restatement of existence and uniqueness of Teichmuller maps between two points in Teichmuller space. The Teichmuller map between two marked Riemann surfaces $X$ and $Y$ gives a quadratic differential on $X$. Stretching along the vertical foliation and contracting along the horizontal one gives the Teichmuller geodesic between $X$ and $Y$.

2) False. There are Teichmuller geodesics which do not have a well-defined limit at infinity in Thurston's compactification. An example is given in the reference by ThiKu. One can nonetheless study their limit sets in the boundary.

3)False. The quadratic differential associated to a Teichmuller geodesic defines a pair of measured foliations $F_1$ and $F_2$ which together fill the surface, meaning that any simple closed curve has positive intersection number with either $F_1$ or $F_2$. It is easy to build a pair of simple closed curve on a genus two surface which do not fill it.

I will add that there is another difference between geodesics in the hyperbolic plane and in a high genus Teichmuller space. In the latter case, there are examples of distinct Teichmuller rays starting at the same point $X$ which converge to the same point in Thurston's compactification. Examples were first constructed by Masur in the paper "Two boundaries for Teichmuller space".

I hope that it helps!

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This subject is addressed in Kasra Rafi's very nice paper (particularly relevant to point 4).

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