A construction of pseudo-Anosov braids with small normalized entropies

Abstract

Let $b$ be a pseudo-Anosov braid whose permutation has a fixed point and let $M_b$ be the mapping torus by the pseudo-Anosov homeomorphism defined on the genus $0$ fiber $F_b$ associated with $b$. This paper describes a structure of the fibered cone $\mathcal{C}$ of $F$ for $M_b$. We prove that there is a $2$-dimensional subcone ${\mathcal{C}}_0$ contained in the fibered cone $\mathcal{C}$ of $F_b$ such that the fiber $F_a$ for each primitive integral class $a\in {\mathcal{C}}_0$ has genus $0$. We also give a constructive description of the monodromy ${\varphi }_a:F_a\to F_a$ of the fibration on $M_b$ over the circle, and consequently provide a construction of many sequences of pseudo-Anosov braids with small normalized entropies. As an application we prove that the smallest entropy among skew-palindromic 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 $1/g$.

31 pages, 19 figures; Remark 6.2 and Lemma 6.3 are added

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Cited by in corpus (2)

• Goeritz groups of bridge decompositions
• On hyperbolic surface bundles over the circle as branched double covers of the $3$-sphere