Geometry & Topology

The centered dual and the maximal injectivity radius of hyperbolic surfaces

Jason DeBlois

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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 rg1,2 such that the set of closed genus-g hyperbolic surfaces with maximal injectivity radius at least r is compact if and only if r > rg1,2. 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

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

Received: 5 September 2013
Revised: 19 March 2014
Accepted: 15 June 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52C15: Packing and covering in $2$ dimensions [See also 05B40, 11H31] 57M50: Geometric structures on low-dimensional manifolds

hyperbolic surface injectivity radius packing Delaunay


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.

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