A uniform bound for geodesic periods of eigenfunctions on hyperbolic surfaces

We consider periods along closed geodesics and along geodesic circles for eigenfunctions of the Laplace-Beltrami operator on a compact hyperbolic Riemann surface. We obtain uniform bounds for such periods as the corresponding eigenvalue tends to infinity. We use methods from the theory of automorphic functions and, in particular, the uniqueness of the corresponding invariant functionals on irreducible unitary representations of PGL₂(ℝ).