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In this article, we constructively prove that on a surface S with genus g ≥ 2, there exit maximal geodesic laminations with 7g − 7, . . . , 9g − 9 leaves. Thus, S can have ideal cell-decompositions (i.e., S can be (ideally) triangulated by maximal geodesic laminations) with 7g − 7, . . . , 9g − 9 (ideal) 1-cells. Once there is a triangulation for a compact surface, the Euler characteristic for the surface can be calculated as the alternating sum F−E+V, where F, E, and V denote the number of faces, edges, and vertices, respectively. We also prove that the same formula holds for the ideal celldecompositions.

IDEAL CELL-DECOMPOSITIONS FOR A HYPERBOLIC SURFACE AND EULER CHARACTERISTIC