By Alexander Ziwet, Louis Allen Hopkins

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More recently, the Fenchel-Nielsen pa rameters have been brought to new life through Thurston's work on hyper bolic manifolds (Thurston [1], Fathi-Laudenbach-Poenaru [1]). The construction is based on the pasting of geodesic hexagons. Let a, b, c be arbitrary positive real numbers. 1 Lemma. There exists a right-angled geodesic hexagon in the hyper bolic plane with pairwise non-adjacent sides of length a, by c. Proof. Consider the perpendicular geodesies fi, a and y in H as in Fig. 1. 1 the same side of j3 as y.

Lift the homotopy between y and c to D. 5). l, §6] 25 Closed Geodesies implies that c intersects R(c). Since c is simple, the unique lifting property implies that for some constant co, R(C(T)) = C{T + co), a contradiction. 7. (iv). Since c is not smooth, y n c = 0 , by (ii). By (iii), y is simple. The proof that y and c bound an embedded annulus is of purely topological nature and is postponed to the Appendix (Proposition A. 11). 6 no longer hold. 8 Example. (Cusps). Let S be a non-compact hyperbolic surface and let W c S be a domain which is isometric to the surface (1) ]-oo, 0 ] x 5 ' = ]-oo, 0] x R/[f i-> f + 1] with the Riemannian metric (2) ds2 = dp2 + e2p dt2 (cf.

The trirectangle is again decomposed into two right-angled triangles with common hypothe nuse. 1 yield the relations cosh r cosh s = coshp! , p2) = cosh r cosh(p2 - s) = cosh r cosh p2 cosh s - cosh r sinh p2 sinh s - coshpj coshp2 cosh(f2 - tx) - sinhp! sinhp2. O We return to the trirectangle of Fig. 1. If side a grows continuously, the vertex at angle

ab, (p^O. 1(vi) or by a glance at Fig.