Volume 21, issue 6 (2021)

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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 $\frac{n}{2}$. We show that the diameter of those random surfaces is asymptotic to $2logn$ in probability as $n\to \infty$.

Keywords
diameter, hyperbolic surfaces, random surfaces
Primary: 57M50
Secondary: 05C80