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The action of the Mapping Class Group on the fundamental group of the complement of a finite subset of a Riemann surface of positive genus

Luca Da Col

TL;DR

This paper delivers a concrete description of how the mapping class group $M(g,n)$ acts on the fundamental group $\pi_1(T_{g,n})$ of a genus $g$ surface with $n$ marked points by expressing the images of generators as outer automorphisms. It builds on the pure mapping class group $P(g,n)$ with a detailed generating set $\mathcal{H}_{g,n}$ (via Dehn twists) and extends to $M(g,n)$ using half-twists $\omega_i$, describing their action on the standard generators $\hat{\alpha}_i,\hat{\beta}_i,\hat{\gamma}_j$ of $\pi_1(T_{g,n})$ and providing explicit formulas. The main results are encapsulated in tables and proved by explicit Dehn-twist computations (sections 4.1–4.8), including the identification of auxiliary loops $\sigma$, $\lambda_i$, and $\mu_{2i,2i+2}$ that express conjugation effects. The work generalizes known low-genus descriptions and enables algorithmic analysis of topological types for group actions on higher-genus surfaces, contributing to the deformation-type classification in related literature. Overall, it provides a complete, constructive framework for understanding $M(g,n)$-actions on $\pi_1(T_{g,n})$ through explicit outer automorphisms.

Abstract

We describe the action of the mapping class group $M(g,n)$ on the fundamental group of $T_{g,n}$, a compact orientable topological surface of positive genus $g$ with $n$ marked points. This is achieved by computing the image of the generators of $M(g,n)$ as outer automorphisms of the fundamental group.

The action of the Mapping Class Group on the fundamental group of the complement of a finite subset of a Riemann surface of positive genus

TL;DR

This paper delivers a concrete description of how the mapping class group acts on the fundamental group of a genus surface with marked points by expressing the images of generators as outer automorphisms. It builds on the pure mapping class group with a detailed generating set (via Dehn twists) and extends to using half-twists , describing their action on the standard generators of and providing explicit formulas. The main results are encapsulated in tables and proved by explicit Dehn-twist computations (sections 4.1–4.8), including the identification of auxiliary loops , , and that express conjugation effects. The work generalizes known low-genus descriptions and enables algorithmic analysis of topological types for group actions on higher-genus surfaces, contributing to the deformation-type classification in related literature. Overall, it provides a complete, constructive framework for understanding -actions on through explicit outer automorphisms.

Abstract

We describe the action of the mapping class group on the fundamental group of , a compact orientable topological surface of positive genus with marked points. This is achieved by computing the image of the generators of as outer automorphisms of the fundamental group.
Paper Structure (13 sections, 38 equations, 17 figures, 3 tables)

This paper contains 13 sections, 38 equations, 17 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Generators of the mapping class groups $P(g,n)$ and $M(g,n)$
  3. The action on the fundamental group $\pi_1(T_{g,n})$
  4. Proof of Theorem \ref{['thm:teoPgn']}
  5. $a_1$
  6. $a_2$
  7. $a_i$, for all $i\in\{2g,\dots,2g+n-2\}$
  8. $b$
  9. $b_i$, for all $i\in\{1,\dots,g-1\}$
  10. $c_{1,2}$
  11. $c_{2i,2i+2}$, for all $i\in\{1,\dots,g-2\}$
  12. $d_1,\dots,d_{n-1}$
  13. Figures

Figures (17)

  • Figure 1: The curves defining the set $\mathcal{H}_{g,n}$.
  • Figure 2: $a_1(\hat{\beta}_1)$.
  • Figure 3: Construction of $\sigma$.
  • Figure 4: $a_2(\hat{\beta}_1)$.
  • Figure 5: $a_2(\hat{\beta}_2)$.
  • ...and 12 more figures

Theorems & Definitions (4)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.4
  • Remark 3.2