# Small dilatation mapping classes coming from the simplest hyperbolic braid

@article{Hironaka2010SmallDM, title={Small dilatation mapping classes coming from the simplest hyperbolic braid}, author={E. Hironaka}, journal={Algebraic \& Geometric Topology}, year={2010}, volume={10}, pages={2041-2060} }

In this paper we study the small dilatation pseudo-Anosov mapping classes arising from fibrations over the circle of a single 3‐manifold, the mapping torus for the “simplest hyperbolic braid”. The dilatations that occur include the minimum dilatations for orientable pseudo-Anosov mapping classes for genus gD2;3;4;5 and 8. We obtain the “Lehmer example” in genus gD 5, and Lanneau and Thiffeault’s conjectural minima in the orientable case for all genus g satisfying gD 2 or 4.mod 6/. Our examples… Expand

#### 51 Citations

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

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Since the set of volumes of hyperbolic 3-manifolds is well ordered, for each fixed g there is a genus-g surface bundle over the circle of minimal volume. Here, we introduce an explicit family of… Expand

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Abstract The main result of this paper is a universal finiteness theorem for the set of all small dilatation pseudo-Anosov homeomorphisms ϕ : S → S , ranging over all surfaces S. More precisely, we… Expand

Quotient families of mapping classes

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Thurston's fibered face theory allows us to partition the set of pseudo-Anosov mapping classes on different compact oriented surfaces into subclasses with related dynamical behavior. This is done via… Expand

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We find the minimum dilatation of pseudo-Anosov homeomorphisms that stabilize an orientable foliation on surfaces of genus three, four, or five, and provide a lower bound for genus six to eight. Our… Expand

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We provide a simple criterion for an element of the mapping class group of a closed surface to have normal closure equal to the whole mapping class group. We apply this to show that every nontrivial… Expand

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Let Fm ⊂ Mod(Sm) be a collection of subsets of the mapping class group of a compact oriented surface Sm of genus gm, where gm is unbounded. We say F = ⋃ m Fm admits asymptotically small dilatations… Expand

NOTES ON PSEUDO-ANOSOVS WITH SMALL DILATATIONS COMING FROM THE MAGIC 3-MANIFOLD (Representation spaces, twisted topological invariants and geometric structures of 3-manifolds : RIMS合宿型セミナー報告集)

- Mathematics
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Let N be the exterior of the 3 chain link C3 (Figure 1) in the three sphere S. Gordon and Wu called N the magic manifold, because they found that N has many interesting non-hyperbolic fillings and… Expand

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

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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… Expand

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Since the set of volumes of hyperbolic 3-manifolds is well ordered, for each fixed g there is a genus-g surface bundle over the circle of minimal volume. Here, we introduce an explicit family of… Expand

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

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- 2010

We denote by δg (resp. δ + ), the minimal dilatation for pseudo-Anosovs (resp. pseudo-Anosovs with orientable invariant foliations) on a closed surface of genus g. This paper concerns the… Expand

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We find the minimum dilatation of pseudo-Anosov homeomorphisms that stabilize an orientable foliation on surfaces of genus three, four, or five, and provide a lower bound for genus six to eight. Our… Expand

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