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Minimal Generation of Mapping Class Groups: A Survey of the Orientable Case

Tulin Altunoz, Mehmetcik Pamuk, Oguz Yildiz

TL;DR

This survey chronicles the progress on minimal generation of orientable mapping class groups, tracing a path from Dehn-twist–based presentations to modern torsion- and involution-generated frameworks. It emphasizes the roles of punctures and boundaries via Birman sequences and half-twists, surveys landmark results such as Wajnryb's two-generator theorem and Korkmaz's three-involutions for large genus, and highlights a new bound showing that for even $p\geq 8$, $\mathrm{Mod}(\Sigma_{13,p})$ is generated by three involutions. By linking generating sets to Teichmüller theory, moduli spaces, and 3-manifold topology, the chapter demonstrates both theoretical economy and practical utility for computations and constructions in low-dimensional topology. It concludes with open questions in low-genus and boundary cases and outlines directions toward uniform presentations and extensions to broader settings.

Abstract

The mapping class group of an orientable surface, which records its symmetries up to isotopy, plays a central role in low-dimensional topology. This chapter explores the foundational problem of determining minimal generating sets for these groups. We chart the development of this area from classical results involving Dehn twist generators to more recent breakthroughs showing that mapping class groups can be generated by just two elements, pairs of torsion elements, or a small collection of involutions. This chapter contains a discussion of the most current results for punctured surfaces, including a new improvement showing that for an even number of punctures $p\geq 8$ the group $\mathrm{Mod}(Σ_{13,p})$ is generated by three involutions. Throughout, we highlight the rich interplay between the algebraic features of these generating sets and the underlying geometric structures they encode. The chapter aims to provide a comprehensive account of the pursuit of algebraic and geometric efficiency within one of topology's most intricate and influential groups.

Minimal Generation of Mapping Class Groups: A Survey of the Orientable Case

TL;DR

This survey chronicles the progress on minimal generation of orientable mapping class groups, tracing a path from Dehn-twist–based presentations to modern torsion- and involution-generated frameworks. It emphasizes the roles of punctures and boundaries via Birman sequences and half-twists, surveys landmark results such as Wajnryb's two-generator theorem and Korkmaz's three-involutions for large genus, and highlights a new bound showing that for even , is generated by three involutions. By linking generating sets to Teichmüller theory, moduli spaces, and 3-manifold topology, the chapter demonstrates both theoretical economy and practical utility for computations and constructions in low-dimensional topology. It concludes with open questions in low-genus and boundary cases and outlines directions toward uniform presentations and extensions to broader settings.

Abstract

The mapping class group of an orientable surface, which records its symmetries up to isotopy, plays a central role in low-dimensional topology. This chapter explores the foundational problem of determining minimal generating sets for these groups. We chart the development of this area from classical results involving Dehn twist generators to more recent breakthroughs showing that mapping class groups can be generated by just two elements, pairs of torsion elements, or a small collection of involutions. This chapter contains a discussion of the most current results for punctured surfaces, including a new improvement showing that for an even number of punctures the group is generated by three involutions. Throughout, we highlight the rich interplay between the algebraic features of these generating sets and the underlying geometric structures they encode. The chapter aims to provide a comprehensive account of the pursuit of algebraic and geometric efficiency within one of topology's most intricate and influential groups.

Paper Structure

This paper contains 21 sections, 16 theorems, 30 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Orientable Surfaces
  3. Background and Classical Generators
  4. Dehn Twists: Fundamental Generators
  5. Algebraic Relations
  6. Elementary Mapping Class Groups
  7. Minimal Generation of $\mathop{\mathrm{Mod}}\nolimits(\Sigma_g)$
  8. GENERATION BY TORSION ELEMENTS
  9. Involution Generators
  10. Surfaces with Punctures
  11. The Birman Exact Sequence and Point-Pushing Maps
  12. Generators for the Full Mapping Class Group
  13. Generators for the Pure Mapping Class Group
  14. The Case of a Single Puncture
  15. The Case of Multiple Punctures
  16. ...and 6 more sections

Key Result

Theorem 2.4

There exists a bijection:

Figures (10)

  • Figure 1: The curves in the lantern relation $t_a t_b t_c t_d = t_x t_y t_z$.
  • Figure 2: The curves in the two-holed torus relation $(t_a t_b t_c)^4 = t_d t_e$.
  • Figure 3: The curves in the one-holed torus relation $(t_a t_b)^6 = t_d$.
  • Figure 4: Humphries' generating set
  • Figure 6: The rotations $\rho_1$ and $\rho_2$ in Kassabov's generating set
  • ...and 5 more figures

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.5: Curve Mapping Criterion
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • Remark 2.11
  • ...and 22 more