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 tessellation 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 π : H 2 → F and let S = π −1 (S). The Voronoi tessellation V of H 2 determined by S is a cell complex structure where each x ∈ S determines a polygonal 2-cell V x defined by:x ∈ S, y ∈ S − {x}}, and each point of V (0) is equidistant from at least 3 points of S (see Section 1). The geometric dual to an edge V x ∩ V y of V is the geodesic arc γ xy in H 2 joining x to y, and the set of geometric duals to edges of V is the edge set of the Delaunay tessellation P determined by S. The covering action of π 1 F on H 2 leaves V and P invariant, and these descend to the tessellations V and P of F . P and V are dual in the sense that their edge sets are canonically bijective, as is the vertex set of each with the face set of the other. However, in some cases an edge e = V x ∩ V y of V is not centered (see Definition 3.1): int e does not intersect the geometric dual γ xy ⊂ P to e. We will regard this as a pathology of P , and "fix" it with the centered dual decomposition. Before we outline this construction, here is a sample application of our methods: Theorem 0.1. Let r β = d β /2 > 0, where cosh d β is the real root of x 3 − 14x 2 − 15x − 4. The Delaunay tessellation of a closed, orientable hyperbolic surface F of genus 2 determined by {x} has all edges centered if F has injectivity radius r ≥ r β at x. It is a triangulation unless r = r β and each edge has length d β , in which case it has a single quadrilateral 2-cell.The numerical value of r β is roughly 1.7006, whereas Boröczky's theorem [1] implies a universal upper bound of r α ∼ = 1.7191 on the injectivity radius of a genus-2 hyperbolic surface at a point x (see Lemma 2.3). Example 2.2 describes a surface F α with injectivity radius r α Partially supported by NSF grant DMS-1007175.

The centered dual and the maximal injectivity radius of hyperbolic surfaces

“…The main tool is a version of the "centered dual complex" that we introduced earlier, a coarsening of the Delaunay complex. In particular, we bound the area of a compact centered dual two-cell below given lower bounds on its side lengths.This paper analyzes the "centered dual complex" of a locally finite subset S of H 2 , first introduced in our prior preprint [5], and applies it to describe the maximal injectivity radius of hyperbolic surfaces. The centered dual complex is a cell decomposition with vertex set S and totally geodesic edges.…”

“…This paper analyzes the "centered dual complex" of a locally finite subset S of H 2 , first introduced in our prior preprint [5], and applies it to describe the maximal injectivity radius of hyperbolic surfaces. The centered dual complex is a cell decomposition with vertex set S and totally geodesic edges.…”

“…The centered dual complex of S is defined in Section 2. This runs parallel to Section 3 of [5], but the definitions are modified to accommodate non-compact Voronoi edges. The fact that motivates our definition is that among cyclic polygons in H 2 , increasing the length of an edge increases area if and only if that edge is not the longest of a non-centered polygon, see [7].…”

“…Centered dual 2-cells are not determined by their edge lengths, but the set of possible centered dual two-cells with a given combinatorics and edge length collection is parametrized by a compact admissible space. This is defined in Section 3.2, which parallels Section 5 of [5].…”