By Conforto F.

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L, §7] 29 The Fenchel-Nielsen Parameters Boundary geodesies which come together must have the same length. On the resulting surface S we obtain 3g - 3 closed geodesies which correspond to the 3g - 3 edges of G. , €3g_3 of these geodesies may be prescribed arbitrarily. In addition to these lengths, we may prescribe twist parameters a 1 ? 3. This yields a (6g - 6)-dimensional space STG of examples of compact hyperbolic surfaces of genus g, modelled over the graph G. 7 we shall see that 2TG is a natural model of the so-called Teichmuller space.

Right-angled pentagons). , coshc = cothacoth/?. Proof. 7(iii) - (v) in the particular case 7 = nl2. 5 Lemma. Let a and b be any positive real numbers satisfying sinh a sinh/? > 1. Then there exists a unique right-angled geodesic pentagon with two consecutive sides of lengths a and b. Proof, a and b are the consecutive orthogonal sides of either a trirectangle, a limiting trirectangle with a vertex at infinity, or a right-angled pentagon. 3), the first two cases are excluded. 6). 2, §4 Hexagons If we paste two right-angled pentagons together along a common side r, we obtain a right-angled hexagon.

Sm are obtained by cutting F open along C. It is not difficult to see, for example, that if C is a set of disjoint simple closed geodesies on a hyperbolic surface 5 then S can be cut open along C. In a similar way we shall frequently cut open surfaces along sets of piecewise geodesic curves. 4 The Universal Covering Every complete unbordered surface of constant curvature - 1 is universally covered by the hyperbolic plane (Cheeger-Ebin [1] or Klingenberg [1, 2]). In this section we adapt this to the complete hyperbolic surfaces with boundary.