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

January 2011

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), .
(doi:10.1016/j.topol.2010.10.007).

## 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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## More information

Submitted date: 10 September 2009

Published date: January 2011

Keywords:
systoles, filling sets of curves

Organisations:
Pure Mathematics

## Identifiers

Local EPrints ID: 68643

URI: http://eprints.soton.ac.uk/id/eprint/68643

PURE UUID: 7ed6c4af-0b3c-43c2-9aeb-48c9f357cb26

## Catalogue record

Date deposited: 14 Sep 2009

Last modified: 08 Jan 2022 02:43

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## Contributors

Author:
Hugo Parlier

Author:
Alexandra Pettet

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