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Simple infinite presentations for the mapping class group of a compact non-orientable surface

Ryoma Kobayashi

Abstract

Omori and the author have given an infinite presentation for the mapping class group of a compact non-orientable surface. In this paper, we give more simple infinite presentations for this group.

Simple infinite presentations for the mapping class group of a compact non-orientable surface

Abstract

Omori and the author have given an infinite presentation for the mapping class group of a compact non-orientable surface. In this paper, we give more simple infinite presentations for this group.

Paper Structure

This paper contains 5 sections, 6 theorems, 10 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main-thm-1']}
  3. Proof of Theorem \ref{['main-thm-2']}
  4. Proof of Theorems \ref{['main-thm-3']} and \ref{['main-thm-4']}
  5. Problems on presentations for $\mathcal{M}(N_{g,n})$

Key Result

Theorem 1.1

For $g\geq1$ and $n\geq0$, $\mathcal{M}(N_{g,n})$ admits a presentation with a generating set $\mathcal{T}\cup\mathcal{Y}$. The defining relations are

Figures (5)

  • Figure 1: A model of a non-orientable surface $N_{g,n}$.
  • Figure 2:
  • Figure 3: Simple closed curves $c_1,\dots,c_k$, $c_0$, $c_0^\prime$ and $d_1,\dots,d_7$ with arrows of a surface.
  • Figure 4: Simple closed curves $\mu$, $\alpha$, $\beta$, $\gamma$, $\delta$$A$, $B$, $C$, $D$ and $E$ of $N_{g,n}$.
  • Figure 5:

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: KO2
  • Corollary 2.2
  • proof
  • Remark 3.1