Pants Decompositions of Random Surfaces

Larry Guth; Hugo Parlier; Robert Young

doi:10.1007/s00039-011-0131-x

Pants Decompositions of Random Surfaces

Larry Guth, Hugo Parlier, Robert Young

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Our goal is to show, in two different contexts, that "random" surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus g for which any pants decomposition requires curves of total length at least g^7/6-ε. Moreover, we prove that this bound holds for most metrics in the moduli space of hyperbolic metrics equipped with the Weil-Petersson volume form. We then consider surfaces obtained by randomly gluing euclidean triangles (with unit side length) together and show that these surfaces have the same property.

Original language	English (US)
Pages (from-to)	1069-1090
Number of pages	22
Journal	Geometric and Functional Analysis
Volume	21
Issue number	5
DOIs	https://doi.org/10.1007/s00039-011-0131-x
State	Published - Oct 2011

Keywords

Bers' constants
Riemann surfaces
Teichmüller and moduli spaces
simple closed geodesics

ASJC Scopus subject areas

Analysis
Geometry and Topology

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10.1007/s00039-011-0131-x

Cite this

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title = "Pants Decompositions of Random Surfaces",

abstract = "Our goal is to show, in two different contexts, that {"}random{"} surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus g for which any pants decomposition requires curves of total length at least g7/6-ε. Moreover, we prove that this bound holds for most metrics in the moduli space of hyperbolic metrics equipped with the Weil-Petersson volume form. We then consider surfaces obtained by randomly gluing euclidean triangles (with unit side length) together and show that these surfaces have the same property.",

keywords = "Bers' constants, Riemann surfaces, Teichm{\"u}ller and moduli spaces, simple closed geodesics",

author = "Larry Guth and Hugo Parlier and Robert Young",

note = "Funding Information: Keywords and phrases: Riemann surfaces, simple closed geodesics, Bers{\textquoteright} constants, Teichm{\"u}ller and moduli spaces 2010 Mathematics Subject Classification: Main: 32G15, Secondary: 53C20, 57M50, 60C05 Research of H.P. supported by Swiss National Science Foundation grant number PP00P2 128557.",

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doi = "10.1007/s00039-011-0131-x",

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AU - Guth, Larry

AU - Parlier, Hugo

AU - Young, Robert

N1 - Funding Information: Keywords and phrases: Riemann surfaces, simple closed geodesics, Bers’ constants, Teichmüller and moduli spaces 2010 Mathematics Subject Classification: Main: 32G15, Secondary: 53C20, 57M50, 60C05 Research of H.P. supported by Swiss National Science Foundation grant number PP00P2 128557.

PY - 2011/10

Y1 - 2011/10

N2 - Our goal is to show, in two different contexts, that "random" surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus g for which any pants decomposition requires curves of total length at least g7/6-ε. Moreover, we prove that this bound holds for most metrics in the moduli space of hyperbolic metrics equipped with the Weil-Petersson volume form. We then consider surfaces obtained by randomly gluing euclidean triangles (with unit side length) together and show that these surfaces have the same property.

AB - Our goal is to show, in two different contexts, that "random" surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus g for which any pants decomposition requires curves of total length at least g7/6-ε. Moreover, we prove that this bound holds for most metrics in the moduli space of hyperbolic metrics equipped with the Weil-Petersson volume form. We then consider surfaces obtained by randomly gluing euclidean triangles (with unit side length) together and show that these surfaces have the same property.

KW - Bers' constants

KW - Riemann surfaces

KW - Teichmüller and moduli spaces

KW - simple closed geodesics

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JF - Geometric and Functional Analysis

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Pants Decompositions of Random Surfaces

Abstract

Keywords

ASJC Scopus subject areas

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