Let M be a complete hyperbolic surface of infinite area. Assuming that its
fundamental group is finitely generated and has critical exponent bigger than
1/2, we obtain an effective equidistribution of closed horocycles in the unit
tangent bundle of M. This extends a result of Sarnak in 1981 for surfaces of
finite area. We use this result to prove an orbital counting statement in
sectors for thin subgroups, with a uniform error term for all congruence
subgroups. This has an application in studying almost prime Pythagorean triples
in the Affine linear sieve.