Algebraic & Geometric Topology

A family of pseudo-Anosov braids with small dilatation

Eriko Hironaka and Eiko Kin

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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.

Article information

Algebr. Geom. Topol., Volume 6, Number 2 (2006), 699-738.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces 57M50: Geometric structures on low-dimensional manifolds

pseudo-Anosov braid train track dilatation Salem–Boyd sequences fibered links Smale horseshoe map


Hironaka, Eriko; Kin, Eiko. A family of pseudo-Anosov braids with small dilatation. Algebr. Geom. Topol. 6 (2006), no. 2, 699--738. doi:10.2140/agt.2006.6.699.

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