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In Introduction to Arithmetic Groups by Dave Morris, I read the comment

"Since a single closed surface has closed geodesics of only countably many different lengths..."

which in context is presented as a well-known general fact about closed surfaces, and no reference is given. I am not familiar with this result. Does it have a name? Is there a place I can read about it, or a simple reason why it must be true?

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It's been a while (well, 20 years) so I won't be able to fill in all the details, but the following should allow you to figure this out. Maybe someone who is still working in the area can provide an easier path to the result.

The result relies on

  1. the (in fact well known, at least to differential geometers) fact that a curve $c$ is a geodesic if and only if it's arc length parametrization is a critical point of the energy functional $E(c) =\frac{1}{2}\int ||c^\prime(t)||^2 \, dt$ and that in this case
  2. the energy is the same as the length and that
  3. the energy functional, if defined on a suitable loop space of $H^{1,2}$ maps, satisfies condition (C) of Palais and Smale

A proof of 3. can be found in Klingenbergs "Lectures on closed geodesic", Theorem 1.4.7, in the setup you'll need for the statement in your question.

The claim you are after is then a consequence of 3. -- you may look it up in Palais and Terng, "Critical Point Theory and Submanifold Geometry", 9.4.8, Corrollary 2 which states that the number of critical values of such a functional in a closed bounded interval is finite. So the energy values of critical points are at most countable, but because of 2. this implies the same claim for the length.

The sources are those I used to work with, today there may be better sources which are easier to read or better accessible.

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