## Algebraic & Geometric Topology

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

#### Abstract

This paper concerns the set $ℳ̂$ 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. Let $N(r)$ be the manifold obtained from $N$ by Dehn filling one cusp along the slope $r∈ℚ$. We prove that for each $g$ (resp. $g≢0(mod6)$), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of $ℳ̂$ defined on a closed surface $Σg$ of genus $g$ is achieved by the monodromy of some $Σg$–bundle over the circle obtained from $N(3−2)$ or $N(1−2)$ by Dehn filling both cusps. These minimizers are the same ones identified by Hironaka, Aaber and Dunfield, Kin and Takasawa independently. In the case $g≡6(mod12)$ we find a new family of pseudo-Anosovs defined on $Σg$ with orientable invariant foliations obtained from $N(−6)$ or $N(4)$ by Dehn filling both cusps. We prove that if $δg+$ is the minimal dilatation of pseudo-Anosovs with orientable invariant foliations defined on $Σg$, then

$limsup g ≡ 6 ( mod 1 2 ) , g → ∞ g log δ g + ≤ 2 log δ ( D 5 ) ≈ 1 . 0 8 7 0 ,$

where $δ(Dn)$ 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 $δ1,n$ is the minimal dilatation of pseudo-Anosovs on a genus $1$ surface with $n$ punctures, then

$limsup n → ∞ n log δ 1 , n ≤ 2 log δ ( D 4 ) ≈ 1 . 6 6 2 8 .$

#### Article information

Source
Algebr. Geom. Topol., Volume 13, Number 6 (2013), 3537-3602.

Dates
Revised: 15 February 2013
Accepted: 19 February 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715740

Digital Object Identifier
doi:10.2140/agt.2013.13.3537

Mathematical Reviews number (MathSciNet)
MR3248741

Zentralblatt MATH identifier
1306.37042

#### Citation

Kin, Eiko; Kojima, Sadayoshi; Takasawa, Mitsuhiko. Minimal dilatations of pseudo-Anosovs generated by the magic $3$–manifold and their asymptotic behavior. Algebr. Geom. Topol. 13 (2013), no. 6, 3537--3602. doi:10.2140/agt.2013.13.3537. https://projecteuclid.org/euclid.agt/1513715740

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