Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Coarsely bounded generating sets for mapping class groups of infinite-type surfaces

Thomas Hill, Sanghoon Kwak, Rebecca Rechkin

TL;DR

The paper addresses the problem of identifying and illustrating $CB$-generating sets for mapping class groups of infinite-type surfaces, clarifying the role of end-space structure in coarse geometry. It surveys Mann–Rafi's $CB$ classification and provides explicit constructions of $CB$-generating sets for surfaces satisfying their hypotheses, including a finite-type core $K$ and finite collections of shifts and half-twists. It also presents a range of examples and non-examples, showing how tameness, limit-type end spaces, and infinite rank obstruct $CB$-generation and when local $CB$ can occur without global $CB$. The work advances understanding of coarse geometric properties of big mapping class groups and offers a blueprint for similar analyses in other big groups.

Abstract

Mann and Rafi's seminal work initiated the study of the coarse geometry of big mapping class groups. Specifically, they construct coarsely bounded (CB) generating sets for mapping class groups of a large class of infinite-type surfaces. In this expository note, we illustrate examples of surfaces whose mapping class groups admit such generating sets, as well as those that do not, with the goal of exploring the context of Mann--Rafi's hypotheses.

Coarsely bounded generating sets for mapping class groups of infinite-type surfaces

TL;DR

The paper addresses the problem of identifying and illustrating -generating sets for mapping class groups of infinite-type surfaces, clarifying the role of end-space structure in coarse geometry. It surveys Mann–Rafi's classification and provides explicit constructions of -generating sets for surfaces satisfying their hypotheses, including a finite-type core and finite collections of shifts and half-twists. It also presents a range of examples and non-examples, showing how tameness, limit-type end spaces, and infinite rank obstruct -generation and when local can occur without global . The work advances understanding of coarse geometric properties of big mapping class groups and offers a blueprint for similar analyses in other big groups.

Abstract

Mann and Rafi's seminal work initiated the study of the coarse geometry of big mapping class groups. Specifically, they construct coarsely bounded (CB) generating sets for mapping class groups of a large class of infinite-type surfaces. In this expository note, we illustrate examples of surfaces whose mapping class groups admit such generating sets, as well as those that do not, with the goal of exploring the context of Mann--Rafi's hypotheses.
Paper Structure (10 sections, 11 theorems, 7 equations, 16 figures, 1 table)

This paper contains 10 sections, 11 theorems, 7 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Infinite-type surfaces
  4. Coarse geometry of topological groups
  5. Mann--Rafi's CB classification
  6. CB-generating set for $\mathop{\mathrm{Map}}\nolimits(S)$
  7. Examples
  8. Connecting finiteness of $H$, $F$, and $T$ to \ref{['thm:CBgen']}
  9. Non-tame examples
  10. Summary

Key Result

Theorem 2.2

Let $S$ and $S'$ be two surfaces of infinite type. Then $S$ is homeomorphic to $S'$ if and only if By $\cong$ between the 4-tuples we mean that $S$ and $S'$ have the same number of boundary components: $b(S) = b(S')$; the same number of genus $g(S) = g(S')$, (possibly infinite); and there exists a homeomorphism of pairs meaning there is a homeomorphism $E(S) \to E(S')$ whose restriction to $E_G(

Figures (16)

  • Figure 1: These two surfaces are homeomorphic and are referred to as the Loch Ness monster surface. On the left, it is clear there is only one way to approach infinity in the surface. However, on the right, the apparently "different ways" to move toward infinity are identified in the one-point compactification of $\mathbb{R}^2$.
  • Figure 2: A surface $S$ with four end types. Here, $x \prec y \sim y' \prec z$, and $v$ is incomparable with all other ends of $S$.
  • Figure 3: Illustration of an end space of limit type. Notice the elements of $E(z_n)$ overlaps $U_n$, but are not entirely contained in $U_n$, while $E(z_n) \cap U_n$'s limit into $X$.
  • Figure 4: Illustration of an end space of a surface with infinite rank mapping class group. Notice $E(z_n)$ has at least two accumulation points, one of which is contained in $U$ and another of which is not.
  • Figure 5: An illustration of the end space $D$ for \ref{['exa:nontame_basic']}. The maximal ends are $z$ and $\{z_n\}_{n \ge 1}$, which are pairwise incomparable to one another. $D$ is not tame as the end $z$ is not stable.
  • ...and 11 more figures

Theorems & Definitions (33)

  • Definition 2.1
  • Theorem 2.2: Richards1963OnTCkerekjarto1923vorlesungen
  • Definition 2.3: mann2023large
  • Proposition 2.4: mann2023large
  • Example 2.5
  • Definition 2.6: rosendal2013global
  • Proposition 2.7
  • proof
  • Definition 2.8: mann2023large
  • Definition 2.9: mann2023large
  • ...and 23 more