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2006 A family of pseudo-Anosov braids with small dilatation
Eriko Hironaka, Eiko Kin
Algebr. Geom. Topol. 6(2): 699-738 (2006). DOI: 10.2140/agt.2006.6.699

Abstract

This paper describes a family of pseudo-Anosov braids with small dilatation. The smallest dilatations occurring for braids with 3,4 and 5 strands appear in this family. A pseudo-Anosov braid with 2g+1 strands determines a hyperelliptic mapping class with the same dilatation on a genus–g surface. Penner showed that logarithms of least dilatations of pseudo-Anosov maps on a genus–g surface grow asymptotically with the genus like 1g, and gave explicit examples of mapping classes with dilatations bounded above by log11g. Bauer later improved this bound to log6g. The braids in this paper give rise to mapping classes with dilatations bounded above by log(2+3)g. They show that least dilatations for hyperelliptic mapping classes have the same asymptotic behavior as for general mapping classes on genus–g surfaces.

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Eriko Hironaka. Eiko Kin. "A family of pseudo-Anosov braids with small dilatation." Algebr. Geom. Topol. 6 (2) 699 - 738, 2006. https://doi.org/10.2140/agt.2006.6.699

Information

Received: 23 July 2005; Revised: 13 April 2006; Accepted: 26 April 2006; Published: 2006
First available in Project Euclid: 20 December 2017

zbMATH: 1126.37014
MathSciNet: MR2240913
Digital Object Identifier: 10.2140/agt.2006.6.699

Subjects:
Primary: 37E30, 57M50

Rights: Copyright © 2006 Mathematical Sciences Publishers

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