Communications in Analysis and Geometry

Volume 19 (2011)

Number 4

Pseudo-Anosov braids with small entropy and the magic $3$-manifold

Pages: 705 – 758

DOI: http://dx.doi.org/10.4310/CAG.2011.v19.n4.a3

Authors

Eiko Kin (Department of Mathematical and Computing Sciences, Tokyo Institute of Technology)

Mitsuhiko Takasawa (Department of Mathematical and Computing Sciences, Tokyo Institute of Technology)

Abstract

We consider a surface bundle over the circle, the so-calledmagic manifold $M$. We determine homology classes whoseminimal representatives are genus $0$ fiber surfaces for$M$, and describe their monodromies by braids. Among thoseclasses whose representatives have $n$ punctures for each$n$, we decide which one realizes the minimal entropy. Weshow that for each $n \ge 9$ (resp. $n= 3,4,5,7,8$), thereexists a pseudo-Anosov homeomorphism $\Phi_n: D_n\rightarrow D_n$ with the smallest known entropy (resp. thesmallest entropy) which occurs as the monodromy on an$n$-punctured disk fiber for the Dehn filling of $M$. Apseudo-Anosov homeomorphism $\Phi_6: D_6 \rightarrow D_6$with the smallest entropy occurs as the monodromy on a$6$-punctured disk fiber for $M$.

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