Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The word problem for the mapping class group in quasi-linear time

Mark C. Bell, Saul Schleimer

TL;DR

This work resolves the word problem for the mapping class group of a compact surface in time $O(n\log^3(n))$, by reducing the general case to a fixed topological setup and then employing a divide-and-conquer pipeline inspired by Dynnikov and half-GCD ideas. Central to the method are $\Delta$-coordinates derived from a pants-based cuff/dual/double-dual system, Dynnikov-type matrices, and a suite of combinatorial moves on tight pairs of train tracks that reduce complexity while preserving intersection data. A coarse, interval-based refinement (interval-weighted train tracks) enables quasi-linear time execution of move sequences, yielding subroutines for fast intersection, curve shortening, and a fast word-parsing pipeline, all with explicit $O()$ bounds and dependence on $|\chi(S)|$. The results extend to corollaries about braid groups, finite-order detection, and related decision problems, representing a significant improvement over prior quadratic-time algorithms and advancing practical computation in the mapping class group. The work highlights a robust interaction between topological decompositions, combinatorial train-track dynamics, and fast integer arithmetic to achieve near-linear complexity.

Abstract

We give an $O(n \log^3(n))$-time algorithm for the word problem in the mapping class group of a compact surface.

The word problem for the mapping class group in quasi-linear time

TL;DR

This work resolves the word problem for the mapping class group of a compact surface in time , by reducing the general case to a fixed topological setup and then employing a divide-and-conquer pipeline inspired by Dynnikov and half-GCD ideas. Central to the method are -coordinates derived from a pants-based cuff/dual/double-dual system, Dynnikov-type matrices, and a suite of combinatorial moves on tight pairs of train tracks that reduce complexity while preserving intersection data. A coarse, interval-based refinement (interval-weighted train tracks) enables quasi-linear time execution of move sequences, yielding subroutines for fast intersection, curve shortening, and a fast word-parsing pipeline, all with explicit bounds and dependence on . The results extend to corollaries about braid groups, finite-order detection, and related decision problems, representing a significant improvement over prior quadratic-time algorithms and advancing practical computation in the mapping class group. The work highlights a robust interaction between topological decompositions, combinatorial train-track dynamics, and fast integer arithmetic to achieve near-linear complexity.

Abstract

We give an -time algorithm for the word problem in the mapping class group of a compact surface.

Paper Structure

This paper contains 33 sections, 31 theorems, 47 equations, 10 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Reduction to a special case
  3. Corollaries
  4. History and other work
  5. Background
  6. Matrices and further reduction
  7. Curves, pants decompositions, and coordinates
  8. Cuffs, dual curves, and double duals
  9. Curve coordinates
  10. Dynnikov's matrices for mapping classes
  11. Train tracks
  12. Index
  13. Train tracks
  14. Weightings
  15. Pairs of train tracks
  16. ...and 18 more sections

Key Result

Theorem 1.1

[theorem]WordProblem There is an algorithm that solves the word problem for $\mathop{\mathrm{MCG}}\nolimits(S)$ in $O(n \log^3(n))$ time. Furthermore, the implied constants are bounded by a fixed polynomial in $|\chi(S)|$.

Figures (10)

  • Figure 3.1: The cuff $\alpha$, its dual $\beta$, and their double dual $\gamma$.
  • Figure 4.1: A switch
  • Figure 4.2: Neighbourhoods of points of $\sigma \cap \tau$ (up to reflection). Here $\sigma$ is shown in red, $\tau$ in blue and $\sigma \cap \tau$ in black.
  • Figure 4.3: Legal high index regions (up to reflection and interchanging $\sigma$ and $\tau$). Here $\sigma$ is shown in red, $\tau$ in blue, $\sigma \cap \tau$ in black and unknown sections of $\sigma \cup \tau$ in dotted gray.
  • Figure 4.4: The branch $b$ can pass through $Q$ forming an I, J, C or S.
  • ...and 5 more figures

Theorems & Definitions (80)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Corollary 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • ...and 70 more