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Mapping class groups with the Rokhlin property

Justin Lanier, Nicholas G. Vlamis

Abstract

We classify the connected orientable 2-manifolds whose mapping class groups have a dense conjugacy class. We also show that the mapping class group of a connected orientable 2-manifold has a comeager conjugacy class if and only if the mapping class group is trivial.

Mapping class groups with the Rokhlin property

Abstract

We classify the connected orientable 2-manifolds whose mapping class groups have a dense conjugacy class. We also show that the mapping class group of a connected orientable 2-manifold has a comeager conjugacy class if and only if the mapping class group is trivial.

Paper Structure

This paper contains 20 sections, 22 theorems, 26 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polish groups and the joint embedding property
  4. The classification of surfaces
  5. Mapping class groups and curves on surfaces
  6. Mapping class groups and their topology
  7. Curve graphs
  8. Subsurface projections
  9. Alexander systems
  10. Ordering ends and a partition of surfaces
  11. Notational conventions
  12. Compact non-displaceable subsurfaces
  13. Self-similar end space
  14. Unique maximal end
  15. Explicit examples of dense conjugacy classes
  16. ...and 5 more sections

Key Result

Corollary 1.1

Let $S$ be a connected orientable non-compact 2-manifold of either zero or infinite genus. If the end space of $S$ is self-similar with a unique maximal end, then every homomorphism from $\mathop{\mathrm{MCG}}\nolimits(S)$ to a cm-slender group is trivial.

Figures (5)

  • Figure 1: The Loch Ness monster surface, the flute surface, and the Cantor tree surface.
  • Figure 2: Two realizations of the Loch Ness monster surface.
  • Figure 3: A realization of the Loch Ness monster surface $L$ produced by identifying $\Sigma$ and the countably many compact surfaces $R_n$ along pairs of boundary components.
  • Figure 4: The 2-manifold $S$ (top). The end space $\mathscr E$ of $S$ (bottom).
  • Figure 5: The curves and subsurfaces used in the proof of Theorem \ref{['thm:meager']} (top). The three scenarios arising in the definition of the elements $g_j \in \mathop{\mathrm{MCG}}\nolimits(\Sigma_j)$ for $j \in \{2, \ldots, n\}$ (bottom).

Theorems & Definitions (41)

  • Corollary 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 3.1
  • ...and 31 more