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Susumu Hirose

Publications and source records attributed to Susumu Hirose.

At least 19 recordsLinked to original sources
arXiv2025

On generating mapping class groups by pseudo-Anosov elements

Wajnryb proved that the mapping class group of a closed oriented surface is generated by two elements. We proved that the mapping class group is generated by two pseudo-Anosov elements. In particular, if the genus is greater than or equal to nine, we can take the generators to two conjugate pseudo-Anosov elements with arbitrarily large dilatations. Another result we prove is that the mapping class group is generated by two conjugate reducible but not periodic elements if the genus is greater than or equal to eight. We also give similar results to the first and third results for the hyperelliptic mapping class group when the genus is greater than or equal to one.

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arXiv2023

Volumes of fibered 2-fold branched covers of 3-manifolds

We prove that for any closed, connected, oriented 3-manifold M, there exists an infinite family of 2-fold branched covers of M that are hyperbolic 3-manifolds and surface bundles over the circle with arbitrarily large volume.

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arXiv2022

Braids, entropies and fibered 2-fold branched covers of 3-manifolds

It is proved by Sakuma and Brooks that any closed orientable $3$-manifold with a Heegaard splitting of genus $g$ admits a $2$-fold branched cover that is a hyperbolic $3$-manifold and a genus $g$ surface bundle over the circle. This paper concerns entropy of pseudo-Anosov monodromies for hyperbolic fibered $3$-manifolds. We prove that there exist infinitely many closed orientable $3$-manifolds $M$ such that the minimal entropy over all hyperbolic, genus $g$ surface bundles over the circle as $2$-fold branched covers of the $3$-manifold $M$ is comparable to $1/g$.

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arXiv2022

Finite presentations for the balanced superelliptic mapping class groups

The balanced superelliptic mapping class group is the normalizer of the transformation group of the balanced superelliptic covering space in the mapping class group of the total surface. We give finite presentations for the balanced superelliptic mapping class groups of closed surfaces, surfaces with one marked point, and surfaces with one boundary component. To give these presentations, we construct finite presentations for corresponding liftable mapping class groups in a different generating set from Ghaswala-Winarski's presentation in \cite{Ghaswala-Winarski1}.

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arXiv2020

Goeritz groups of bridge decompositions

For a bridge decomposition of a link in the $3$-sphere, we define the Goeritz group to be the group of isotopy classes of orientation-preserving homeomorphisms of the $3$-sphere that preserve each of the bridge sphere and link setwise. After describing basic properties of this group, we discuss the asymptotic behavior of the minimal pseudo-Anosov entropies. This gives an application to the asymptotic behavior of the minimal entropies for the original Goeritz groups of Heegaard splittings of the $3$-sphere and the real projective space.

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arXiv2020

A construction of pseudo-Anosov braids with small normalized entropies

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 $ ϕ_a: F_a \rightarrow 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 $1/g$.

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arXiv2019

On hyperbolic surface bundles over the circle as branched double covers of the $3$-sphere

The branched virtual fibering theorem by Sakuma states that every closed orientable $3$-manifold with a Heegaard surface of genus $g$ has a branched double cover which is a genus $g$ surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minimal entropy over all hyperbolic, genus $g$ surface bundles as branched double covers of the $3$-sphere behaves like 1/$g$. We also give an alternative construction of surface bundles over the circle in Sakuma's theorem when closed $3$-manifolds are branched double covers of the $3$-sphere branched over links. A feature of surface bundles coming from our construction is that the monodromies can be read off the braids obtained from the links as the branched set.

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arXiv2017

The asymptotic behavior of the minimal pseudo-Anosov dilatations in the hyperelliptic handlebody groups

We consider the hyperelliptic handlebody group on a closed surface of genus $g$. This is the subgroup of the mapping class group on a closed surface of genus $g$ consisting of isotopy classes of homeomorphisms on the surface that commute with some fixed hyperelliptic involution and that extend to homeomorphisms on the handlebody. We prove that the logarithm of the minimal dilatation (i.e, the minimal entropy) of all pseudo-Anosov elements in the hyperelliptic handlebody group of genus $g$ is comparable to $1/g$. This means that the asymptotic behavior of the minimal pseudo-Anosov dilatation of the subgroup of genus $g$ in question is the same as that of the ambient mapping class group of genus $g$. We also determine finite presentations of the hyperelliptic handlebody groups.

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arXiv2017

On topological classification of finite cyclic actions on bordered surfaces

In [Tohoku Math. J. 62 (2010), 45--53] the second author showed that, except for a few cases, the order $N$ of a cyclic group of self-homeomorphisms of a closed orientable topological surface $S_g$ of genus $g \geq 2$ determines the group up to a topological conjugation, provided that $N\geq 3g$. The first author et al. undertook in [Collect. Math. 67 (2016), 415--429] a more general problem of topological classification of such group actions for $N>2(g-1)$. In [Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A. Mat. (RACSAM) 110 (2016), 303--320] we considered the analogous problem for closed non-orientable surfaces, and in [J. Pure Appl. Algebra 220 (2016) 465--481] - the problem of classification of cyclic actions generated by an orientation reversing self-homeomorphism. The present paper, in which we deal with topological classification of actions on bordered surfaces of finite cyclic groups of order $N>p-1$, where $p$ is the algebraic genus of the surface, completes our project of topological classification of "large" cyclic actions on compact surfaces. We apply obtained results to solve the problem of uniqueness of the actions realising the solutions of the so called minimum genus and maximum order problems for bordered surfaces.

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arXiv2015

A normal generating set for the Torelli group of a non-orientable closed surface

For a closed surface $S$, its Torelli group $\mathcal{I}(S)$ is the subgroup of the mapping class group of $S$ consisting of elements acting trivially on $H_1(S;\mathbb{Z})$. When $S$ is orientable, a generating set for $\mathcal{I}(S)$ is known. In this paper, we give a normal generating set of $\mathcal{I}(N_g)$ for $g\geq4$, where $N_g$ is a genus-$g$ non-orientable closed surface.

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arXiv2015

A uniqueness of periodic maps on surfaces

Kulkarni showed that, if g is greater than 3, a periodic map on an oriented surface S_g of genus g with order more than or equal to 4g is uniquely determined by its order, up to conjugation and power. In this paper, we show that, if g is greater than 30, the same phenomenon happens for periodic maps on the surfaces with orders more than 8g/3 and, for any integer N, there is g > N such that there are periodic maps of S_g of order 8g/3 which are not conjugate up to power each other. Moreover, as a byproduct of our argument, we provide a short proof of Wiman's classical theorem: the maximal order of periodic maps of S_g is 4g+2.

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arXiv2013

Seifert surfaces in open books, and pass moves on links

The flat plumbing basket presentation of a link is introduced by Furihata, Hirasawa and Kobayashi. In this paper, we show that the pass-equivalence and an equivalence introduced by using the flat plumbing basket presentation are the same relation. Furthermore, we obtain an evaluation of the minimal number of bands used for the flat pluming basket presentation for a knot coming from the degree of the Alexander polynomial and the three genus of the knot.

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