Geodesic
A family of pseudo-Anosov braids with small dilatation
Hironaka, Eriko1 ; Kin, Eiko2
1Department of Mathematics, Florida State University, Tallahassee FL 32306-4510, USA
2Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-45 Oh-okayama, Meguro-ku, Tokyo 152-8552, Japan
Algebraic and Geometric Topology, Tome 6 (2006) no. 2, pp. 699-738
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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 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.

DOI : 10.2140/agt.2006.6.699
Keywords: pseudo-Anosov, braid, train track, dilatation, Salem–Boyd sequences, fibered links, Smale horseshoe map

Hironaka, Eriko  1   ; Kin, Eiko  2

1 Department of Mathematics, Florida State University, Tallahassee FL 32306-4510, USA
2 Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-45 Oh-okayama, Meguro-ku, Tokyo 152-8552, Japan 
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Hironaka, Eriko; Kin, Eiko. A family of pseudo-Anosov braids with small dilatation. Algebraic and Geometric Topology, Tome 6 (2006) no. 2, pp. 699-738. doi: 10.2140/agt.2006.6.699

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