The centered dual and the maximal injectivity radius of hyperbolic surfaces

Abstract

We give sharp upper bounds on the maximal injectivity radius of finite-area hyperbolic surfaces and use them, for each $g\ge 2$, to identify a constant $r_{g-1,2}$ such 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. In particular, we bound the area of a compact centered dual two-cell below given lower bounds on its side lengths.