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Big mapping class groups: an overview

Javier Aramayona, Nicholas G. Vlamis

Abstract

We survey recent developments on mapping class groups of surfaces of infinite topological type.

Big mapping class groups: an overview

Abstract

We survey recent developments on mapping class groups of surfaces of infinite topological type.

Paper Structure

This paper contains 40 sections, 51 theorems, 34 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Surfaces and their classification
  4. Some important examples
  5. Arcs and curves
  6. Mapping class group
  7. Several natural subgroups
  8. Pure mapping class group
  9. Compactly supported mapping class group
  10. Torelli group
  11. Modular groups
  12. Two important results
  13. Alexander method
  14. Automorphisms of the curve graph
  15. Topological aspects
  16. ...and 25 more sections

Key Result

Theorem 2.1

For any surface $S$, the space $\operatorname{Ends}(S)$ is totally disconnected, second countable, and compact. In particular, $\operatorname{Ends}(S)$ is homeomorphic to a closed subset of a Cantor set.

Figures (4)

  • Figure 1: Infinite-type deer, by Juan Pablo Díaz González, UNAM
  • Figure 2: From left to right: Loch Ness monster surface, Jacob's ladder surface, and the blooming Cantor tree surface.
  • Figure 3: The circles are identified vertically to obtain $\Sigma$.
  • Figure 4: The rigid structure on $S$.

Theorems & Definitions (54)

  • Theorem 2.1
  • Theorem 2.2: Classification, KeRichards
  • Theorem 2.3: Realization, Richards
  • Proposition 2.4
  • Theorem 3.1: HMV
  • Theorem 3.2
  • Proposition 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4
  • ...and 44 more