ABSTRACT. This paper determines those Riemann surfaces on which each nonsimple closed geodesic has a parabolic intersection-that is, an intersection in the form of a loop enclosing a puncture or a deleted disk. An application is made characterizing the simple closed geodesic on H/T{3) in terms of the Markov spectrum.The thrust of the situation is this: If we call loops about punctures or deleted disks boundary curves, then if the surface has "little" topology, each nonsimple closed geodesic must contain a boundary curve. But if there is "enough" topology, there are nonsimple closed geodesies not containing boundary curves.1. Let 72 be a Riemann surface of genus g with k punctures and d disks removed; k, d > 0, k + d > 0. In this paper we consider the closed geodesies on 72.The method used is to represent 72 as a quotient, 72 = 77/r, where H = {z = x + iy: y > 0} and T is a fuchsian group acting on 77. Let it : 77 -► 77/r be the projection map. We assume (77, n) is an unramified covering, so T has no elliptic elements. For most of this paper we shall assume k > 0 and d = 0; the remaining cases are considered in §5. Each hyperbolic axis projects to a closed geodesic in 72. Each closed geodesic a in 72 lifts to a conjugacy class [a] in T and under known conditions [a] is hyperbolic.Let 7 G T be hyperbolic with axis A-,. From now on we assume 7 primitive; i.e., 7 generates the stabilizer of A1. It is easily seen that (A^) is simple iff A1 fl ßA1 = 0 for all ß G T -(7).We write A~, A ßA^ to mean that A~¡ fl ßA1 = {z} for some z in 77. Thus (1.1) ""(-A-y) is nonsimple (= self-intersecting) iff A1 A ßA1 for some ß E V -(7).Since aA^ = Aaia-i, conjugate elements are both simple or both nonsimple. We use the phrase a hyperbolic 7 is simple (nonsimple) to mean n(A1) is simple (nonsimple).It is readily checked that if 7 is nonsimple, we can choose ß in (1.1) to be hyperbolic; indeed, we may replace ß by /?7n for large n. However, it is not always possible to choose ß to be parabolic, and in this connection we prove the following result.