CURVES, SURFACES, AND HYPERBOLIC GEOMETRY 23 the basepoint for 1 (S) to some point on . The resulting element of 1 (S) is well defined only up to conjugacy. By a slight abuse of notation we will denote this element of 1 (S) by as well. There is a bijective correspondence: Nontrivial free Nontrivial homotopy classes of oriented conjugacy classes closed curves in S in 1 (S) An element g of a group G is primitive if there does not exist any h G so that g = h k , where |k| > 1. The property of being primitive is a conjugacy class invariant. In particular, it makes sense to say that a closed curve in a surface is primitive. A closed curve in S is a multiple if it is a map S 1 S that factors ×n through the map S 1 - S 1 for n > 1. In other words, a curve is a multiple if it "runs around" another curve multiple times. If a closed curve in S is a multiple, then no element of the corresponding conjugacy class in 1 (S) is primitive. Let p : S S be any covering space. By a lift of a closed curve to S we will always mean the image of a lift R S of the map , where