[Paper Review] Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex

Eiko Kin, Hyunshik Shin|arXiv (Cornell University)|Jul 19, 2017

Geometric and Algebraic Topology11 references5 citations

TL;DR

This paper establishes that in hyperbolic fibered 3-manifolds with first Betti number ≥ 2, there exists a sequence of pseudo-Anosov monodromies on surfaces with Euler characteristic tending to infinity, whose asymptotic translation lengths on the curve complex decay precisely as $1/|\chi(R_n)|^2$. This construction reproves and extends Gadre–Tsai's result on closed surfaces and establishes the same asymptotic scaling for the hyperelliptic mapping class group and hyperelliptic handlebody group.

ABSTRACT

Let $M$ be a hyperbolic fibered 3-manifold with $b_1(M) \geq 2$ and let $S$ be a fiber with pseudo-Anosov monodromy $ψ$. We show that there exists a sequence $(R_n, ψ_n)$ of fibers and monodromies contained in the fibered cone of $(S,ψ)$ such that the asymptotic translation length of $ψ_n$ on the curve complex $\mathcal{C}(R_n)$ behaves asymptotically like $1/|χ(R_n)|^2$. As applications, we can reprove the previous result by Gadre--Tsai that the minimal asymptotic translation length of a closed surface of genus $g$ asymptotically behaves like $1/g^2$. We also show that this also holds for the cases of hyperelliptic mapping class group and hyperelliptic handlebody group.

Motivation & Objective

• To establish the asymptotic behavior of minimal asymptotic translation lengths of pseudo-Anosov maps on the curve complex in fibered 3-manifolds with $b_1(M) \geq 2$.
• To reprove and extend Gadre–Tsai's result that $L_{\mathcal{C}}(\mathrm{Mod}(S_g)) \asymp 1/g^2$ using a geometric construction from 3-manifold topology.
• To show that the same asymptotic scaling $1/|\chi|^2$ holds for the hyperelliptic mapping class group and the hyperelliptic handlebody group.
• To construct explicit sequences of pseudo-Anosov mapping classes with translation lengths decaying as $1/|\chi(R_n)|^2$ via lifts in $\mathbb{Z}$-covers of surfaces.

Proposed method

• Lift the monodromy $\psi$ of a fiber $S$ in a fibered 3-manifold $M$ to a $\mathbb{Z}$-cover $\widetilde{S}$ corresponding to a fixed primitive cohomology class $\xi_0$.
• Construct a sequence of surfaces $R_n = \widetilde{S}/\langle h^n \widetilde{\psi} \rangle$, where $h$ is the deck transformation, and show $|\chi(R_n)| \asymp n$.
• Prove that the monodromy $\psi_n$ on $R_n$ is pseudo-Anosov and that its asymptotic translation length satisfies $\ell_{\mathcal{C}}(\psi_n) \asymp 1/|\chi(R_n)|^2$.
• Use the nesting lemma and train track techniques from Masur–Minsky and Bestvina–Handel to bound translation lengths from below.
• Establish an upper bound by constructing a curve $\alpha$ such that $d_{\mathcal{C}}(\alpha, \psi_n^m(\alpha)) = 1$ for some $m \asymp |\chi(R_n)|^2$, implying $\ell_{\mathcal{C}}(\psi_n) \leq C / |\chi(R_n)|^2$.
• Apply the construction to the Hilden group and handlebody group by lifting braids and using the mapping torus of a 5-braid to generate the required fibered 3-manifold structure.

Experimental results

Research questions

• RQ1 Can the asymptotic scaling $L_{\mathcal{C}}(\mathrm{Mod}(S_g)) \asymp 1/g^2$ be rederived using geometric constructions in 3-manifold topology?
• RQ2 Does the $1/g^2$ scaling hold for subgroups of the mapping class group, such as the hyperelliptic mapping class group and hyperelliptic handlebody group?
• RQ3 What is the asymptotic behavior of the minimal asymptotic translation length in fibered 3-manifolds with $b_1(M) \geq 2$?
• RQ4 Can a sequence of pseudo-Anosov maps be constructed in a fibered 3-manifold such that $\ell_{\mathcal{C}}(\psi_n) \asymp 1/|\chi(R_n)|^2$?
• RQ5 How does the structure of the $\mathbb{Z}$-cover and the action of $h^n \widetilde{\psi}$ on the curve complex relate to translation length decay?

Key findings

• For any hyperbolic fibered 3-manifold $M$ with $b_1(M) \geq 2$, there exists a sequence of fibers $(R_n, \psi_n)$ such that $|\chi(R_n)| \asymp n$ and $\ell_{\mathcal{C}}(\psi_n) \asymp 1/|\chi(R_n)|^2$.
• The construction reproves Gadre–Tsai's result that $L_{\mathcal{C}}(\mathrm{Mod}(S_g)) \asymp 1/g^2$ using a geometric, 3-manifold-based method.
• The same asymptotic scaling $\ell_{\mathcal{C}}(\psi_n) \asymp 1/|\chi(R_n)|^2$ holds for the hyperelliptic mapping class group $\mathcal{H}(S_g)$, with $L_{\mathcal{C}}(\mathcal{H}(S_g)) \leq 1/(g^2 - 2g - 1)$ for $g \geq 3$.
• The result extends to the hyperelliptic handlebody group $\mathcal{H}(\mathbb{H}_g)$, where $L_{\mathcal{C}}(\mathcal{H}(\mathbb{H}_g)) \leq C/g^2$ for some constant $C > 0$.
• The upper bound is established by constructing a curve $\alpha$ with $d_{\mathcal{C}}(\alpha, \psi_n^m(\alpha)) = 1$ for $m \asymp |\chi(R_n)|^2$, implying $\ell_{\mathcal{C}}(\psi_n) \leq C / |\chi(R_n)|^2$.
• The construction relies on lifting a 5-braid to a $\mathbb{Z}$-cover and taking quotients by $h^n \widetilde{\psi}$, yielding surfaces with punctures and monodromies satisfying the desired asymptotic behavior.