Algebraic & Geometric Topology

A family of pseudo-Anosov braids with small dilatation

Eriko Hironaka and Eiko Kin

Full-text: Open access

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.

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

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

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

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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