Twisted Rokhlin property for mapping class groups

2023/12/05 by Pravin Kumar, Kumar, Pravin, Apeksha Sanghi +3
Mathematics · #54H11 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Primary 57K20 #Rings, Modules, and Algebras #Secondary 20E36

paper · pdf · doi:10.48550/arxiv.2312.02539

openalex publication_date 2023/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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_∞-property.

Related

• Twisted Conjugacy in Big Mapping Class Groups
• Twisted conjugacy classes in Symplectic groups, Mapping class groups and Braid groups(including an Appendix written with Francois Dahmani)
• Twisted conjugacy classes in twisted Chevalley groups
• Twisted conjugacy classes in special and general linear groups
• Big mapping class groups and the co-Hopfian property