Quadratic-time computations for pseudo-Anosov mapping classes

TL;DR

This work solves the Nielsen–Thurston problem for pseudo-Anosov elements by presenting a quadratic-time algorithm that, given a word in a fixed mapping class generating set, outputs a Nielsen–Thurston package $(\tau,D)$ for the element. The core idea is to model the action of the mapping class group on the space of measured foliations MF$(S)$ as an action on an integral cone complex, within fixed train-track coordinates, and to guarantee that an iterate $f^Q$ lands in a Nielsen–Thurston eigenregion where the unstable foliation emerges as the leading eigenvector of the corresponding sink matrix $D$. The main technical contributions are the forcing lemma, the fitting lemma, and the development of slope- and horizontality-based bounds that control how foliations, train tracks, and rectangles interact under iteration; these yield effective, terminating procedures to extract stretch factors and foliations. The results advance practical computation in the Nielsen–Thurston classification, provide tools for translating dynamics on MF$(S)$ into concrete train-track data, and offer potential algorithms for conjugacy and translation-length problems in the mapping class group. Overall, the paper delivers a rigorous, implementable framework for obtaining canonical dynamical data of pseudo-Anosov elements in time quadratic in input length, with explicit constants tied to the topology of the surface.

Abstract

We give a quadratic-time algorithm to compute the stretch factor and the invariant measured foliations for a pseudo-Anosov element of the mapping class group. As input, the algorithm accepts a word (in any given finite generating set for the mapping class group) representing a pseudo-Anosov mapping class, and the length of the word is our measure of complexity for the input. The output is a train track and an integer matrix where the stretch factor is the largest real eigenvalue and the unstable foliation is given by the corresponding eigenvector. This is the first algorithm to compute stretch factors and measured foliations that is known to terminate in sub-exponential time.

Figures (11)

• Figure 1: A geometric realization of the integral cone complex $Y_\Delta$
• Figure 2: Train tracks giving a cell structure on $\mathop{\mathrm{MF}}\nolimits(S_{0,4})$
• Figure 3: In the proof of Theorem \ref{['thm:main']}, we choose $Q$ so $\mathcal{H}_f(f^Q(c))$ is very large, we find a coordinate train track $\tau_i$ carrying, $\mathcal{F}_h$, and we choose an $f$-invariant $\nu$ carried by $\tau_i$ with $\mathcal{H}_f(\nu)$ small
• Figure 4: The integral cone complex structure $Y$ on $\mathop{\mathrm{MF}}\nolimits(\mathop{\mathrm{D}}\nolimits_3)$ and the common refinement of $Y_\Gamma$
• Figure 5: A part of the computation of the linear transformation $V_{12} \to V_2$ induced by $\sigma_1$

Theorems & Definitions (48)

• Theorem 1.1
• Corollary 1.2
• Proposition 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• Lemma 3.5
• proof
• proof : Proof of Lemma \ref{['lemma:global-disparity']}