Journal of the
Korean Mathematical Society JKMS

In this article, we constructively prove that on a surface $S$ with genus $g\geq 2,$ there exit maximal geodesic laminations with $7g-7,\ldots, 9g-9$ leaves. Thus, $S$ can have $ideal$ $cell$-$decompositions$ (i.e., $S$ can be (ideally) triangulated by maximal geodesic laminations) with $7g-7,\ldots, 9g-9$ (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 proved that the same formula holds for the ideal cell-decompositions.

Keywords: ideal cell-decomposition, geodesic lamination, Euler characteristic

MSC numbers: 57M99