Tessellations of hyperbolic surfaces

Abstract. A finite subset S of a closed hyperbolic surface F canonically determines a centered dual decomposition of F: a cell structure with vertex set S, geodesic edges, and 2-cells that are unions of the corresponding Delaunay polygons. Unlike a Delaunay polygon, a centered dual 2-cell Q is not determined by its collection of edge lengths; but together with its combinatorics, these determine an admissible space parametrizing geometric possibilities for the Delaunay cells comprising Q. We illustrate its application by using the centered dual decomposition to extract combinatorial information about the Delaunay tessellation among certain genus-2 surfaces, and with this relate injectivity radius to covering radius here. A finite subset S of a closed hyperbolic surface F canonically determines a Voronoi tes-sellation V and Delaunay tessellation P, polygonal decompositions of F that are dual in a certain sense. Let us briefly outline this construction. Fix a locally isometric universal covering pi: H2 → F and let S ̃ = pi−1(S). The Voronoi tessellation V ̃ of H2 determined by S ̃ is a cell complex structure where each x ∈ S determines a polygonal 2-cell Vx defined by: Vx = {p ∈ H2 | d(p, x) ≤ d(p, y) for each y ∈ S − {x}}(0.0.1) Then V ̃ (1)