arXiv subjects

Mitsuhiko Takasawa

Publications and source records attributed to Mitsuhiko Takasawa.

7 recordsLinked to original sources

arXiv2013

The boundary of a fibered face of the magic 3-manifold and the asymptotic behavior of the minimal pseudo-Anosovs dilatations

Let $δ_{g,n}$ be the minimal dilatation of pseudo-Anosovs defined on an orientable surface of genus $g$ with $n$ punctures. Tsai proved that for any fixed $g \ge 2$, the logarithm of the minimal dilatation $\log δ_{g,n}$ is on the order of $\frac{\log n}{n}$. The main result of this paper is that if $2g+1$ is relatively prime to $s$ or $s+1$ for each $0 \le s \le g$, then $$\limsup_{n \to \infty} \frac{n \log δ_{g,n}}{\log n} \le 2.$$

math.GT↗

arXiv2012

Minimal dilatations of pseudo-Anosovs generated by the magic 3-manifold and their asymptotic behavior

This paper concerns the set $\hat{\mathcal{M}}$ of pseudo-Anosovs which occur as monodromies of fibrations on manifolds obtained from the magic 3-manifold $N$ by Dehn filling three cusps with a mild restriction. We prove that for each $g$ (resp. $g \not\equiv 0 \pmod{6}$), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of $\hat{\mathcal{M}}$ defined on a closed surface $\varSigma_g$ of genus $g$ is achieved by the monodromy of some $\varSigma_g$-bundle over the circle obtained from $N(\tfrac{3}{-2})$ or $N(\tfrac{1}{-2})$ by Dehn filling two cusps. These minimizers are the same ones identified by Hironaka, Aaber-Dunfiled, Kin-Takasawa independently. In the case $g \equiv 6 \pmod{12}$ we find a new family of pseudo-Anosovs defined on $\varSigma_g$ with orientable invariant foliations obtained from N(-6) or N(4) by Dehn filling two cusps. We prove that if $δ_g^+$ is the minimal dilatation of pseudo-Anosovs with orientable invariant foliations defined on $\varSigma_g$, then $$ \limsup_{\substack{g \equiv 6 \pmod{12} g \to \infty}} g \log δ^+_g \le 2 \log δ(D_5) \approx 1.0870,$$ where $δ(D_n)$ is the minimal dilatation of pseudo-Anosovs on an $n$-punctured disk. We also study monodromies of fibrations on N(1). We prove that if $δ_{1,n}$ is the minimal dilatation of pseudo-Anosovs on a genus 1 surface with $n$ punctures, then $$ \limsup_{n \to \infty} n \log δ_{1,n} \le 2 \log δ(D_4) \approx 1.6628. $$

math.GT↗

arXiv2011

Pseudo-Anosovs on closed surfaces having small entropy and the Whitehead sister link exterior

Let $δ_g$ be the minimal dilatation for pseudo-Anosovs on a closed surface $Σ_g$ of genus $g$ and let $δ_g^+$ be the minimal dilatation for pseudo-Anosovs on $Σ_g$ with orientable invariant foliations. This paper concerns the pseudo-Anosovs which occur as the monodromies on closed fibers for Dehn fillings of $N(r)$ for each $r \in \{-3/2, -1/2, 2\}$ of the magic manifold $N$. The manifold $N(-3/2)$ is homeomorphic to the Whitehead sister link exterior. We consider the set $Λ_g(r)$ (resp. $Λ_g^+(r)$) which consists of the dilatations of all monodromies (resp. monodromies having orientable invariant foliations) on a closed fiber of genus $g$ for Dehn fillings of $N(r)$, where the fillings are on the boundary slopes of fibers of $N(r)$. Hironaka obtained upper bounds of $δ_g$ and $δ_g^+$ by computing $\min Λ_g(-1/2)$ and $\min Λ^+_g(-1/2)$ respectively. We prove that $\min Λ_g(-3/2)< \min Λ_g(-1/2)$ for $g \equiv 0,1,5,6,7,9 \pmod{10}$ and $\min Λ^+_g(-3/2)< \min Λ^+_g(-1/2)$ for $g \equiv 1,5,7,9 \pmod{10}$. These inequalities improve the previous upper bounds of $δ_g$ and $δ_g^+$ for these $g$. We prove that for each $r \in \{-3/2, -1/2, 2\}$ and each $g \ge 3$, there exists a monodromy $Φ_g(r)$ on a closed fiber of genus $g $ for a Dehn filling of $N(r)$ such that its dilatation $λ(Φ_g(r))$ satisfies $\displaystyle \lim_{g \to \infty} |χ(Σ_g)| \log λ(Φ_g(r)) = 2 \log((3+\sqrt{5})/2)$.

math.GT↗

arXiv2010

Constructions of surface bundles with rank two fundamental groups

We give a construction of hyperbolic 3-manifolds with rank two fundamental groups and report an experimental search to find such manifolds. Our manifolds are all surface bundles over the circle with genus two surface fiber. For the manifolds so obtained, we then examine whether they are of Heegaard genus two or not. As a byproduct, we give an infinite family of fibered knots of genus two in the 3-sphere whose knot groups are of rank two.

math.GT↗

arXiv2010

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

We consider a hyperbolic surface bundle over the circle with the smallest known volume among hyperbolic manifolds having 3 cusps, so called "the magic manifold". We compute the entropy function on the fiber face of the unit ball with respect to the Thurston norm, determine homology classes whose representatives are genus 0 fiber surfaces, and describe their monodromies by braids. Among such homology classes whose representatives have n punctures, we decide which one realizes the minimal entropy. It turns out that all the braids with smallest known entropy are derived from monodromies for such homology classes.

math.GT↗

arXiv2008

Entropy vs volume for pseudo-Anosov maps

We will discuss theoretical and experimental results concerning comparison of entropy of pseudo-Anosov maps and volume of their mapping tori. Recent study of Weil-Petersson geometry of the Teichmüller space tells us that they admit linear inequalities for both sides under some bounded geometry condition. We construct a family of pseudo-Anosov maps which violates one side of inequalities under unbounded geometry setting, present an explicit bounding constant for a punctured torus, and provide several observations based on experiments.

math.GT↗

arXiv2007

An asymptotic behavior of the dilatation for a family of pseudo-Anosov braids

The dilatation of a pseudo-Anosov braid is a conjugacy invariant. In this paper, we study the dilatation of a special family of pseudo-Anosov braids. We prove an inductive formula to compute their dilatation, a monotonicity and an asymptotic behavior of the dilatation for this family of braids. We also give an example of a family of pseudo-Anosov braids with arbitrarily small dilatation such that the mapping torus obtained from such braid has 2 cusps and has an arbitrarily large volume.

math.GT↗