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The diameter of random Belyĭ surfaces

Thomas Budzinski, Nicolas Curien and Bram Petri

Algebraic & Geometric Topology 21 (2021) 2929–2957
Abstract

We determine the asymptotic growth rate of the diameter of the random hyperbolic surfaces constructed by Brooks and Makover (J. Differential Geom. 68 (2004) 121–157). This model consists of a uniform gluing of 2n hyperbolic ideal triangles along their sides followed by a compactification to get a random hyperbolic surface of genus roughly n 2 . We show that the diameter of those random surfaces is asymptotic to 2logn in probability as n .

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Keywords
diameter, hyperbolic surfaces, random surfaces
Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 05C80
References
Publication
Received: 12 November 2019
Revised: 24 September 2020
Accepted: 20 October 2020
Published: 22 November 2021
Authors
Thomas Budzinski
Mathematics Department
University of British Columbia
Vancouver, BC
Canada
Nicolas Curien
Département de Mathématique
Université Paris-Saclay and Institut Universitaire de France
Orsay
France
Bram Petri
Institut de Mathématiques de Jussieu-Paris Rive Gauche
Sorbonne Université
Paris
France