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In torus there exists simple non-closed geodesic. One example is take the irrational slope curves in $\mathbb{R}^2$ and project it down to torus. Is this thing can happen in closed hyperbolic surface i.e.

Does there exists a simple non-closed geodesic in a closed (i.e. compact without boundary) hyperbolic surface?

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    $\begingroup$ I know almost nothing about the subject, but I think the answer is yes, and it is related to laminations. $\endgroup$ Commented May 27, 2013 at 16:30
  • $\begingroup$ @Oliver it is actually related to lamination. But why you think this is true? It will be helpful if you share your idea or point of view. $\endgroup$ Commented May 27, 2013 at 17:05
  • $\begingroup$ I don't understand why you accepted Neal's answer. As Daniel Rust commented, it didn't answer your question, because it didn't show whether the non-closed geodedics could be simple or not. $\endgroup$
    – 23rd
    Commented May 31, 2013 at 11:13
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    $\begingroup$ @Landscape There is another reason. Rust's references explain how you can construct one of these. $\endgroup$ Commented Jun 1, 2013 at 3:51

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Let's call $\Sigma$ a closed hyperbolic surface. Each free homotopy class of curves on a hyperbolic surface has exactly one geodesic representative. Conversely, a closed geodesic gives a free homotopy class. Thus (up to basepoint) there is a bijective correspondence $$\bigg\{ \mbox{ closed geodesics }\bigg\} \longleftrightarrow \pi_1(\Sigma).$$

Since $\pi_1(\Sigma)$ is finitely generated, it is a countable set. In particular, it lifts to a countable set of geodesics in $\mathbb{H}^2$. Any and every geodesic in $\Sigma$ can be obtained by pushing a geodesic of $\mathbb{H}^2$ forward under the covering map, so the set of all geodesics of $\Sigma$ lifts to the set of geodesics of $\mathbb{H}^2$, which is uncountable. So there are many more open geodesics than closed geodesics on $\Sigma$.


Here's a simple example on a non-closed hyperbolic cylinder that illustrates the phenomenon. Let $\Sigma = \mathbb{H}^2/\mathbb{Z}$, where the $\mathbb{Z}$ action is given by $z\mapsto \lambda z$, for some $\lambda > 1$. The single closed geodesic is the image (under quotient) of the $y$-axis. Take any other geodesic which limits to $0$ and push it down under the covering map; it will spiral in from one side of the cylinder and limit to the closed geodesic.

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    $\begingroup$ Only a small point, using this method, can you be sure that at least one geodesic in $\mathbb{H}^2$ maps down to a simple, non-closed geodesic on $\Sigma$? $\endgroup$
    – Dan Rust
    Commented May 30, 2013 at 18:07
  • $\begingroup$ @DanielRust Good good good question. I think something like this might work: If a geodesic $\gamma$ has two lifts which cross, choose a second geodesic $\beta$ which connects two of the limit points of the two lifts and project it. $\endgroup$
    – Neal
    Commented May 30, 2013 at 18:39
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For closed hyperbolic surfaces (with constant negative curvature) the answer is yes. This is shown by exhibiting a transitive geodesic flow on the surface. In this paper by Hedlund, the sufficient property of the flow is called regional transitivity and implies the existence of a simple dense geodesic on the surface $M$. As the geodesic is dense in $M$, it can not be closed.

In the above paper, it is stated in Theorem 3.1 that there are a countable number of periodic geodesics and so in some sense 'most' geodesics on $M$ are non-closed. To actually construct such a geodesic (although not necessarily simple), the standard technique is to use 'symbolic trajectories' which Hedlund hints at in the very last section of the paper. Essentially, you can lift any geodesic on $M$ to its universal cover by the hyperbolic plane and create a bi-infinite sequence associated to the geodesic which tells you which edges of the fundamental regions the geodesic passes through as you 'walk along' the geodesic. Such a sequence is sometimes called a cutting sequence. Admissible cutting sequences can be totally classified by local configuration rules and so it is then just a matter of finding a non-periodic admissible cutting sequences, which will correspond to non-periodic (and hence non-closed) geodesics on the surface $M$.

For an introduction to cutting sequences, and their beautiful links with continued fractions (as well as a direct construction of a non-periodic geodesic on the modular surface) I would strongly recommend reading this wonderful and easy to read article by Caroline Series.

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    $\begingroup$ Thanks. These references are excellent. Thanks again. $\endgroup$ Commented May 30, 2013 at 16:01
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    $\begingroup$ Could you please be more specific to explain how Hedlund's paper implies that there exists a simple dense geodesic? $\endgroup$
    – 23rd
    Commented May 31, 2013 at 11:09
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    $\begingroup$ A simple geodesic on a hyperbolic surface cannot be dense! Hadlund's result only proves that generic geodesics are dense. $\endgroup$ Commented Sep 4, 2013 at 20:46

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