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I'm trying to learn about the Selberg zeta function, but it seems like introductory texts assume more knowledge of Riemannian geometry than I'm comfortable with.

I have some basic questions that someone might be able to help with:

Is the composition of two closed geodesics itself a closed geodesic?

Is composition of geodesics a commutative operation?

Can all geodesics be decomposed into compositions of primitive closed geodesics?

If anyone has comments or references, I would appreciate them.

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    Selberg is not (yet?) in Abel's league, so we still write his last name capitalized :)2010-09-25
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    @Mariano: Dear Mariano, I'm not sure about your first claim, but I agree with the second: Selberg should be capitalized.2010-09-26
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    Dear Simon, Think about composing two great circles on a sphere. Is this typically another great circle?2010-09-26
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    @Matt, heh: I really meant not to underestimate Selberg's value! The league I had in mind was the Uncapitalized League :)2010-09-26
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    @Mariano Quoting Selberg: "I want to make it clear that I never have read in detail Abel’s so-called Paris Memoir, which for a long time disappeared before it was recovered and published long after Abel’s death. But it is that little note, upon which the results of the Paris Memoir rest, which is so extremely elementary. There really is no comparison in the mathematical literature, I think. Such a fundamental and far-reaching theorem proved by so simple and elementary methods—it is pure magic. I cannot imagine anything that quite compares to this."2011-07-30

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