#### Volume 19, issue 2 (2015)

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The centered dual and the maximal injectivity radius of hyperbolic surfaces

### Jason DeBlois

Geometry & Topology 19 (2015) 953–1014
##### 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.

##### Keywords
hyperbolic surface, injectivity radius, packing, Delaunay
##### Mathematical Subject Classification 2010
Primary: 52C15, 57M50