The centered dual and the maximal injectivity radius of hyperbolic surfaces

TL;DR

This paper establishes sharp upper bounds on the maximal injectivity radius of finite-area hyperbolic surfaces, characterizes the compactness of the moduli space based on this radius, and introduces a new geometric tool called the centered dual complex.

Contribution

It introduces the centered dual complex as a new geometric tool and applies it to derive bounds on the injectivity radius of hyperbolic surfaces.

Findings

Sharp upper bounds on maximal injectivity radius for hyperbolic surfaces.

Identification of a critical radius r_{g-1,2} for compactness of moduli space.

Bounded the area of centered dual cells based on side lengths.

Abstract

We give sharp upper bounds on the maximal injectivity radius of finite-area hyperbolic surfaces and use them, for each g at least 2, to identify a constant r_{g-1,2} with the property that the set of closed genus-g hyperbolic surfaces with maximal injectivity radius at least r is compact if and only if r > r_{g-1,2}. The main tool is a version of the "centered dual complex" that we introduced earlier, a coarsening of the Delaunay complex of a locally finite set. In particular, we bound the area of a compact centered dual two-cell below given lower bounds on its side lengths.