Pants decompositions of random surfaces
type: Item
instance of: scholarly article
title: Pants decompositions of random surfaces
zbMATH Open document ID: 1242.32007
author: Larry Guth
author: Hugo Parlier
author: Robert Young
published in: Geometric and Functional Analysis. GAFA
publication date: 2012-01-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.0612
full work available at URL: Geom. Funct. Anal. Vol. 21 2011 .pdf
review text: It is well-known that a compact hyperbolic Riemann surface \(X\) of genus \(g\) can be decomposed as a connected sum of three-holed spheres (usually called pairs of pants). The total pants length of such a decomposition is the sum of the lengths of the curves used for cutting the surface; and the total pants length of \(X\) is the infimum of the total pants length of its decompositions. In this paper the authors show that for any \(\varepsilon>0\) the measure (with respect to the normalized Weil-Petersson volume on the moduli space of hyperbolic surfaces of genus \(g\)) of the set of hyperbolic surfaces of genus \(g\) with total pants length at least \(g^{7/6-\varepsilon}\) tends to~1 as \(g\) tends to~\(+\infty\). The authors also prove a similar result for combinatorial surfaces built by glueing together equilateral triangles to form a closed surface: for any \(\varepsilon>0\) the measure (with respect to the uniform measure on the finite set of combinatorial surfaces obtained by glueing \(N\) triangles) of the set of combinatorial surfaces built with \(N\) triangles and with total pants length at least \(N^{7/6-\varepsilon}\) tends to~1 as \(N\) tends to~\(+\infty\).
reviewed by: Marco Abate
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 60C05
zbMATH DE Number: 6000006
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: simple closed geodesics
zbMATH Keywords: Bers' constant
zbMATH Keywords: Teichmüller spaces
zbMATH Keywords: moduli spaces
zbMATH Keywords: total pants length
zbMATH Keywords: combinatorial surfaces
MaRDI profile type: Publication
OpenAlex ID: W2041307487
arXiv ID: 1011.0612
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DOI: 10.1007/S00039-011-0131-X
description: scientific article