We briefly present some facts from hyperbolic geometry which are helpful in understanding prime geodesics.
It can be shown that this gives a 1-1 correspondence between closed geodesics on Γ\H and hyperbolic conjugacy classes in Γ.
In his 1970 Ph.D. thesis, Grigory Margulis proved a similar result for surfaces of variable negative curvature, while in his 1980 Ph.D. thesis, Peter Sarnak proved an analogue of Chebotarev's density theorem.
See Covering map and Splitting of prime ideals in Galois extensions for more details.
Closed geodesics have been instrumental in studying the eigenvalues of Laplacian operators, arithmetic Fuchsian groups, and Teichmüller spaces.