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Relative shapes of thick subsets of moduli space

Relative shapes of thick subsets of moduli space
Relative shapes of thick subsets of moduli space
A closed hyperbolic surface of genus g ≥ 2 can be decomposed into pairs of pants along shortest closed geodesics and if these curves are sufficiently short (and with lengths uniformly bounded away from 0), then the geometry of the surface is essentially determined by the combinatorics of the pants decomposition. These combinatorics are determined by a trivalent graph, so we call such surfaces trivalent. In this paper, in a first attempt to understand the "shape" of the subset Xg of moduli space consisting of surfaces whose systoles fill, we compare it metrically, asymptotically in g, with the set Yg of trivalent surfaces. As our main result, we find that the set Xg∩Yg is metrically "sparse" in Xg (where we equip Mg with either the Thurston or the Teichmuller metric).
0002-9327
473-498
Anderson, James W.
739c0e33-ef61-4502-a675-575d08ee1a98
Parlier, Hugo
4f1d0bfb-618f-4ca4-881d-bdf6fde4033f
Pettet, Alexandra
0f3e1618-2f05-427e-b9bb-78fb2affb390
Anderson, James W.
739c0e33-ef61-4502-a675-575d08ee1a98
Parlier, Hugo
4f1d0bfb-618f-4ca4-881d-bdf6fde4033f
Pettet, Alexandra
0f3e1618-2f05-427e-b9bb-78fb2affb390

Anderson, James W., Parlier, Hugo and Pettet, Alexandra (2016) Relative shapes of thick subsets of moduli space. American Journal of Mathematics, 138 (2), 473-498. (doi:10.1353/ajm.2016.0010).

Record type: Article

Abstract

A closed hyperbolic surface of genus g ≥ 2 can be decomposed into pairs of pants along shortest closed geodesics and if these curves are sufficiently short (and with lengths uniformly bounded away from 0), then the geometry of the surface is essentially determined by the combinatorics of the pants decomposition. These combinatorics are determined by a trivalent graph, so we call such surfaces trivalent. In this paper, in a first attempt to understand the "shape" of the subset Xg of moduli space consisting of surfaces whose systoles fill, we compare it metrically, asymptotically in g, with the set Yg of trivalent surfaces. As our main result, we find that the set Xg∩Yg is metrically "sparse" in Xg (where we equip Mg with either the Thurston or the Teichmuller metric).

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More information

Submitted date: 26 June 2013
Accepted/In Press date: 19 August 2015
Published date: 2016
Organisations: Pure Mathematics

Identifiers

Local EPrints ID: 354764
URI: http://eprints.soton.ac.uk/id/eprint/354764
ISSN: 0002-9327
PURE UUID: 02230585-a0e4-4bfe-a8c5-61f5a7270c0e
ORCID for James W. Anderson: ORCID iD orcid.org/0000-0002-7849-144X

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Date deposited: 31 Jul 2013 09:07
Last modified: 17 Dec 2019 01:58

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