Some basic questions about the Selberg zeta function 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.
 A: 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.
A: 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
