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Normal generators for mapping class groups

Hyungryul Baik, Dongryul M. Kim

TL;DR

The article surveys the interplay between normal generators in mapping class groups and dynamical invariants, notably translation lengths on Teichmüller space and the curve graph. It centers on Lanier–Margalit’s criterion, which connects small Teichmüller translation lengths of pseudo-Anosov elements to normal generation, and extends these ideas to partly pseudo-Anosov reducibles. It contrasts translation lengths on Teichmüller space (logarithmic in the stretch factor) with those on the curve graph, highlighting results like Penner’s bounds and minimal asymptotics, and presents constructions that show small curve-graph translation lengths do not guarantee normal generation. The piece also raises broad open questions about universal thresholds, the scope of the criterion, and extensions to subgroups such as the handlebody group, guiding future investigations into the normal subgroup structure of mapping class groups.

Abstract

In this expository note, we discuss normal generators for mapping class groups of surfaces. Especially, we focus on the relation between normal generation of a mapping class with its asymptotic translation lengths on the Teichmüller space and the curve graph of the underlying surface. We also discuss several open questions.

Normal generators for mapping class groups

TL;DR

The article surveys the interplay between normal generators in mapping class groups and dynamical invariants, notably translation lengths on Teichmüller space and the curve graph. It centers on Lanier–Margalit’s criterion, which connects small Teichmüller translation lengths of pseudo-Anosov elements to normal generation, and extends these ideas to partly pseudo-Anosov reducibles. It contrasts translation lengths on Teichmüller space (logarithmic in the stretch factor) with those on the curve graph, highlighting results like Penner’s bounds and minimal asymptotics, and presents constructions that show small curve-graph translation lengths do not guarantee normal generation. The piece also raises broad open questions about universal thresholds, the scope of the criterion, and extensions to subgroups such as the handlebody group, guiding future investigations into the normal subgroup structure of mapping class groups.

Abstract

In this expository note, we discuss normal generators for mapping class groups of surfaces. Especially, we focus on the relation between normal generation of a mapping class with its asymptotic translation lengths on the Teichmüller space and the curve graph of the underlying surface. We also discuss several open questions.

Paper Structure

This paper contains 6 sections, 18 theorems, 81 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Mapping class groups
  3. Lanier-Margalit's well-suited criterion and its generalizations
  4. Small translation lengths and normal generation
  5. Asymptotic translation lengths on curve graphs
  6. Further questions

Key Result

Theorem 1.1

Let $f \in \operatorname{Mod}(S_g)$ for $g \geq 3$. Then $f$ is a normal generator if one of the followings holds:

Figures (8)

  • Figure 1: Twist $h$ on an annulus
  • Figure 2: Dehn twist along $c$
  • Figure 3: $c = c_1 \cup c_2 \cup c_3$ and $d = d_1$ fill the surface
  • Figure 4: Finite cyclic covering of degree $g$
  • Figure 5: A separating curve $\beta$ on $S_2$ with $\alpha \cap \beta = \emptyset$
  • ...and 3 more figures

Theorems & Definitions (27)

  • Theorem 1.1: Lanier-Margalit
  • Theorem 1.2: Baik-Kim-Wu
  • Theorem 1.3: Baik-Kim-Wu
  • Definition 2.1: Mapping class group and pure mapping class group
  • Example 2.2: Dehn twists, Dehn_twist
  • Example 2.3: Multitwists
  • Theorem 2.4: Dehn, Lickorish
  • Theorem 2.5: Nielsen-Thurston classification
  • Theorem 2.6: Harer
  • Definition 2.7: Torelli group
  • ...and 17 more