Horocycles on Riemann surfaces

Abstract

By the Collar Theorem, every puncture on a hyperbolic Riemann surface with punctures has a horocyclic neighborhood of area 2 2 . Furthermore two such neighborhoods associated to different punctures are disjoint. This result can be improved if we omit the condition that horocyclic neighborhoods of different punctures must be disjoint. Using arguments of the second author we show, in this paper, that each puncture of a hyperbolic Riemann surface has a horocyclic neighborhood of area 4 4 .