## Algebraic & Geometric Topology

### A family of pseudo-Anosov braids with small dilatation

#### 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 $1∕g$, and gave explicit examples of mapping classes with dilatations bounded above by $log11∕g$. Bauer later improved this bound to $log6∕g$. 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796542

Digital Object Identifier
doi:10.2140/agt.2006.6.699

Mathematical Reviews number (MathSciNet)
MR2240913

Zentralblatt MATH identifier
1126.37014

#### Citation

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. https://projecteuclid.org/euclid.agt/1513796542

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