2006/06/12 by Eriko Hironaka, Eiko Kin · 3 citations
Computer Science · Mathematics · #Botany #Bounded function #Braid #Combinatorics #Genus #Geometric and Algebraic Topology #Geometry #Logarithm #Mapping class group #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Pure mathematics #Surface (topology) #math.GT #msc:37E30 #msc:57M50 #semigroups and automata theory
paper · pdf · doi:10.2140/agt.2006.6.699
published as Algebr. Geom. Topol. 6 (2006) 699-738 · This is the version published by Algebraic & Geometric Topology on 12 June 2006
openalex publication_date 2006/06/12 · arxiv created 2009/04/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 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 + √ 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.