Geodesic Length Functions and Teichm

Given a compact orientable surface , let S(() be the set of iso-topy classes of essential simple closed curves in. We determine a complete set of relations for a function from S(() to R to be the geodesic length function of a hyperbolic metric with geodesic boundary on. As a consequence, the Teichm uller space of hyperbolic metrics with geodesic boundary on is reconstructed from an intrinsic combinatorial structure on S((). This also gives a complete description of the image of Thurston's embedding of the Teichm uller space. x1. Introduction Let = g;r be a compact oriented surface of genus g with r boundary components. The Teichm uller space of isotopy classes of hyperbolic metrics with geodesic boundary on is denoted by T((), and the set of isotopy classes of essential simple closed unoriented curves in is denoted by S = S((). For each m 2 T(() and 2 S((), let l m () be the length of the geodesic representing. An interesting question is to characterize the geodesic length functions among all functions deened on S((). We announce in this note a solution to this question. The solution is expressed in terms of an intrinsic combinatorial structure on S((). Before stating the theorem, let us consider three basic examples which motivate the solution. We denote the isotopy class of a curve s by s], and ft 2 Rjt > ag by R >a. Example 1. If the surface is the three-holed sphere 0;3 , then the set S((0;3) consists of three elements which are the isotopy classes of the three boundary components of the surface. It is well known from the work of Fricke-Klein FK] that any positive function from S(() to R >0 is the geodesic length function of a unique element in the Teichm uller space. For the rest of the note, we introduce the trace function t m () = 2coshl m ()=2 from S(() to R >2. We will deal with the trace function t m instead of l m. Example 2. The surface is the one-holed torus 1;1. Let S 0 be the set fs] 2 Sj s is not homotopic to the boundary @ 1;1 g. It is well known that S 0 is in one-one correspondence with the set of rational numbers Q f1g where the map