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A family of pseudo-Anosov braids with small dilatation

Eriko Hironaka and Eiko Kin

Algebraic & Geometric Topology 6 (2006) 699–738

arXiv: 0904.0594


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.

pseudo-Anosov, braid, train track, dilatation, Salem–Boyd sequences, fibered links, Smale horseshoe map
Mathematical Subject Classification 2000
Primary: 37E30, 57M50
Forward citations
Received: 23 July 2005
Revised: 13 April 2006
Accepted: 26 April 2006
Published: 12 June 2006
Eriko Hironaka
Department of Mathematics
Florida State University
Tallahassee FL 32306-4510
Eiko Kin
Department of Mathematical and Computing Sciences
Tokyo Institute of Technology
2-12-1-W8-45 Oh-okayama
Tokyo 152-8552