Twisted Rokhlin property for mapping class groups

TL;DR

This work introduces the twisted Rokhlin property for topological groups and classifies when big mapping class groups $\mathrm{MCG}(S)$ of connected orientable infinite-type surfaces without boundary have a dense $\phi$-twisted conjugacy class for every automorphism $\phi$. The authors adapt and extend the framework used for the Rokhlin property by Lanier and Vlamis, exploiting end-space structure, curve graphs, and projection techniques to distinguish end-space configurations that allow or forbid the property. They show that $\mathrm{MCG}(S)$ has the twisted Rokhlin property precisely when every compact subsurface is displaceable and $\mathcal{M}(S)$ is a singleton, while in the two-maximal-end, Cantor-end-space, or compact non-displaceable subsurface cases the twisted Rokhlin property fails for all automorphisms. Additionally, every big mapping class group $\mathrm{MCG}(S)$ for connected orientable infinite-type $S$ without boundary has the $R_{\infty}$-property, ensuring infinitely many twisted conjugacy classes for any automorphism. The results enrich the understanding of dynamical properties of big mapping class groups and raise questions about Rokhlin-type phenomena in broader topological-group contexts.

Abstract

In this paper, generalising the idea of the Rokhlin property, we explore the concept of the twisted Rokhlin property of topological groups. A topological group is said to exhibit the twisted Rokhlin property if, for each automorphism $φ$ of the group, there exists a $φ$-twisted conjugacy class that is dense in the group. We provide a complete classification of connected orientable infinite-type surfaces without boundaries whose mapping class groups possess the twisted Rokhlin property. Additionally, we prove that the mapping class groups of the remaining surfaces do not admit any dense $φ$-twisted conjugacy class for any automorphism $φ$. This supplements the recent work of Lanier and Vlamis on the Rokhlin property of big mapping class groups. We also prove that the mapping class group of each connected orientable infinite-type surface without boundary possesses the $R_\infty$-property.

Figures (6)

• Figure 1: Realization of the Loch Ness monster surface.
• Figure 2: Projections of the curve $c$ on $\Sigma$.
• Figure 3: The curve $c$ on $S_{0,6}$ such that $\{ c, \mathfrak{t}(c)\}$ fills $S_{0,6}$.
• Figure 4: The curve $c$ on $S_{g,n}$ such that $\{ c,\mathfrak{t}(c) \}$ fills $S_{g,n}$.
• Figure 5: The curve $c$ such that $\{c, \mathfrak{t}(c)\}$ forms a multicurve.

Theorems & Definitions (56)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition 2.4
• Lemma 2.5
• Lemma 2.6
• proof
• Proposition 2.7