A family of pseudo-Anosov braids with small dilatation

Hironaka, Eriko; Kin, Eiko

doi:10.2140/agt.2006.6.699

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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

DOI: 10.2140/agt.2006.6.699

arXiv: 0904.0594

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 $2 g + 1$ 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 $log 11 ∕ g$ . Bauer later improved this bound to $log 6 ∕ g$ . The braids in this paper give rise to mapping classes with dilatations bounded above by $log (2 + \sqrt{3}) ∕ g$ . They show that least dilatations for hyperelliptic mapping classes have the same asymptotic behavior as for general mapping classes on genus– $g$ surfaces.

Keywords

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

Mathematical Subject Classification 2000

Primary: 37E30, 57M50

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Publication

Received: 23 July 2005

Revised: 13 April 2006

Accepted: 26 April 2006

Published: 12 June 2006

Authors

Eriko Hironaka
Department of Mathematics Florida State University Tallahassee FL 32306-4510 USA

Eiko Kin
Department of Mathematical and Computing Sciences Tokyo Institute of Technology 2-12-1-W8-45 Oh-okayama Meguro-ku Tokyo 152-8552 Japan

Volume 6, issue 2 (2006)