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.
On
http://arxiv.org/abs/math/0407288 you may start your voyage here with 'introduction to the Selberg zeta function' it explain its role for Riemann Hypothesis and a generalization of the Poisson summation formula
http://matwbn.icm.edu.pl/ksiazki/aa/aa91/aa9132.pdf HERE it explain the Zeta function of Selberg in term of the determinant of a certain Laplacian over the surface , also the zeta regularization for determinant is used
for zeta regularization and functional determinants see http://arxiv.org/PS_cache/arxiv/pdf/1108/1108.5659v1.pdf
Hyperbolic geodesic can be run once, twice, three times, every time resulting in a new hyperbolic geodesic formally ..., which admit a multiple of the origninal length.
The primitive one are those, which are not a multiple of any other geodesic.
This can be phrased in terms of generators of the fundamental group, which gives the translation to Fuchsian groups, i.e. $\pi_1(\Gamma \backslash \mathbb{H}) \cong \Gamma$.
Your Riemann surface has no singularieties, iff the generators of $\Gamma$ are in one to one correspondance to primitive geodesics.
So your first question: ... if and only if they are a multiple of the same primitive geodesic. But it is not possible to combine two arbitrary closed geodesics.
2nd question: ... Yes, the topological operation commutes, if it is well defined.
3rd question: ... Yes, every closed geodesic is a multiple of an unique primitive geodesic.
Reference: Iwaniec - Spectral theory of automorphic forms.
Google Fuchsian groups.