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Three perfect mapping class groups

Nicholas G. Vlamis

Abstract

We prove that the mapping class group of a surface obtained from removing a Cantor set from either the 2-sphere, the plane, or the interior of the closed 2-disk has no proper countable-index subgroups. The proof is an application of the automatic continuity of these groups, which was established by Mann. As corollaries, we see that these groups do not contain any proper finite-index subgroups and that each of these groups have trivial abelianization.

Three perfect mapping class groups

Abstract

We prove that the mapping class group of a surface obtained from removing a Cantor set from either the 2-sphere, the plane, or the interior of the closed 2-disk has no proper countable-index subgroups. The proof is an application of the automatic continuity of these groups, which was established by Mann. As corollaries, we see that these groups do not contain any proper finite-index subgroups and that each of these groups have trivial abelianization.

Paper Structure

This paper contains 2 sections, 4 theorems.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1

The mapping class group of a surface obtained by removing a Cantor set from either the 2-sphere, the plane, or the interior of the closed 2-disk does not contain a proper countable-index normal subgroup.

Theorems & Definitions (4)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Corollary 4