## Geometry & Topology

### The centered dual and the maximal injectivity radius of hyperbolic surfaces

Jason DeBlois

#### Abstract

We give sharp upper bounds on the maximal injectivity radius of finite-area hyperbolic surfaces and use them, for each $g ≥ 2$, to identify a constant $rg−1,2$ such that the set of closed genus-$g$ hyperbolic surfaces with maximal injectivity radius at least $r$ is compact if and only if $r > rg−1,2$. 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.

#### Article information

Source
Geom. Topol., Volume 19, Number 2 (2015), 953-1014.

Dates
Revised: 19 March 2014
Accepted: 15 June 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858717

Digital Object Identifier
doi:10.2140/gt.2015.19.953

Mathematical Reviews number (MathSciNet)
MR3336276

Zentralblatt MATH identifier
1330.51007

#### Citation

DeBlois, Jason. The centered dual and the maximal injectivity radius of hyperbolic surfaces. Geom. Topol. 19 (2015), no. 2, 953--1014. doi:10.2140/gt.2015.19.953. https://projecteuclid.org/euclid.gt/1510858717

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