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Small filling sets of curves on a surface

Small filling sets of curves on a surface
Small filling sets of curves on a surface
Consider a set of simple closed curves on a surface of genus g which fill the surface and which pairwise intersect at most once. We show that the asymptotic growth rate of the smallest number in such a set is 2\sqrt{g} as g goes to infinity. More generally, we give a precise asymptotic for filling sets of curves which pairwise intersect at most K times, where K is greater than equal to 1. We then bound from below the cardinality of a filling set of systoles by g/log(g). The topological condition that a set of curves pairwise intersect at most once is thus quite far from the geometric condition that a set of curves can arise as systoles.
systoles, filling sets of curves
84-92
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 (2011) Small filling sets of curves on a surface. Topology and its Applications, 158 (1), 84-92.

Record type: Article

Abstract

Consider a set of simple closed curves on a surface of genus g which fill the surface and which pairwise intersect at most once. We show that the asymptotic growth rate of the smallest number in such a set is 2\sqrt{g} as g goes to infinity. More generally, we give a precise asymptotic for filling sets of curves which pairwise intersect at most K times, where K is greater than equal to 1. We then bound from below the cardinality of a filling set of systoles by g/log(g). The topological condition that a set of curves pairwise intersect at most once is thus quite far from the geometric condition that a set of curves can arise as systoles.

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Submitted date: 10 September 2009
Published date: January 2011
Keywords: systoles, filling sets of curves
Organisations: Pure Mathematics

Identifiers

Local EPrints ID: 68643
URI: https://eprints.soton.ac.uk/id/eprint/68643
PURE UUID: 7ed6c4af-0b3c-43c2-9aeb-48c9f357cb26
ORCID for James W. Anderson: ORCID iD orcid.org/0000-0002-7849-144X

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Date deposited: 14 Sep 2009
Last modified: 06 Jun 2018 13:03

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Contributors

Author: Hugo Parlier
Author: Alexandra Pettet

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