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Any lift α˜ : [0, 1] → H of the loop α satisfies α(0) ˜ = α(1). 28. If we view α and α˜ as curves defined on R such that α(0) = α(1) then we have for all t ˜ . α(t ˜ + 1) = Γα α(t) So there is φ ∈ Conf(H) such that φΓα φ −1 either equals P ± : z → z ± 1 or T : z → e z for some > 0. Replacing the universal cover π with the universal ˜ we may assume that Γα equals one of cover π ◦ φ −1 and replacing α˜ with φ(α) these standard isometries. Hence we have to consider the following two cases (see Fig.

If p, q ∈ G then the geodesic arc connecting p with q is also contained in G. If one or more of the sides bk is replaced by a point on the real line or the point {∞} then G is called a degenerate hexagon (see Fig. 8). The metric gH + on the upper half plane induces a hyperbolic metric on any hexagon. With respect to this metric hexagons always have finite area. 3, [74] pp. 83–85). Consider now a hexagon G ⊂ H with sides a1 , b1 , a2 , b2 , a3 , b3 parameterized on the unit interval [0, 1]. Let now G := {x + iy ∈ C | x − iy ∈ G} be a copy of G in the negative half-plane H − .

4 Annuli We discuss now Riemann surfaces A which are diffeomorphic to (0, 1) × S 1 (open annuli or cylinders). 40 (Hyperbolic cylinders) Consider the geodesic δ(t) = iet in H. Let now γ and γ be geodesics intersecting δ orthogonally at the points ia and ia , respectively, where a < a (see Fig. 5). Parameterize γ and γ with unit speed and with orientations as indicated in Fig. 5. Then the isometry T : z → aa z (with = log(a /a) to be consistent with our previous notation) maps δ onto itself, and it satisfies T (γ (t)) = γ (t).