 

Susumu Hirose and
Eiko Kin
A construction of pseudoAnosov braids with small normalized entropies
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Published: 
May 26, 2020. 
Keywords: 
mapping class groups, pseudoAnosov, dilatation, normalized entropy,
fibered 3manifolds, braid group. 
Subject: 
57M99, 37E30. 


Abstract
Let b be a pseudoAnosov braid whose permutation has a fixed point and let M_{b} be the mapping torus by the pseudoAnosov homeomorphism defined on the genus 0 fiber F_{b} associated with b. We prove that there is a 2dimensional subcone C_{0} contained in the fibered cone C of F_{b} such that the fiber F_{a} for each primitive integral class a ∈ C_{0} has genus 0. We also give a constructive description of the monodromy
φ_{a}: F_{a} → F_{a} of the fibration on M_{b} over the circle, and consequently provide a construction of many sequences of pseudoAnosov braids with small normalized entropies. As an application we prove that the smallest entropy among skewpalindromic braids with n strands is comparable to 1/n, and the smallest entropy among elements of the odd/even spin mapping class groups of genus g is comparable to 1/g.


Acknowledgements
We would like to thank Mitsuhiko Takasawa for helpful conversations and comments. The first author was supported by GrantinAid for Scientific Research (C) (No. 16K05156), Japan Society for the Promotion of Science. The second author was supported by GrantinAid for Scientific Research (C) (No. 18K03299), Japan Society for the Promotion of Science.


Author information
Susumu Hirose:
Department of Mathematics
Faculty of Science and Technology
Tokyo University of Science
Noda, Chiba, 2788510, Japan
hirose_susumu@ma.noda.tus.ac.jp
Eiko Kin:
Department of Mathematics
Graduate School of Science
Osaka University Toyonaka
Osaka 5600043, Japan
kin@math.sci.osakau.ac.jp

