Homeomorphism groups of 2-manifolds with the virtual Rokhlin property

TL;DR

This paper introduces the virtual Rokhlin property for topological groups and classifies 2-manifolds whose homeomorphism and mapping class groups possess this property, advancing understanding of their structure.

Contribution

It defines the virtual Rokhlin property for topological groups and classifies 2-manifolds and their mapping class groups with this property.

Findings

Classification of 2-manifolds with homeomorphism groups having the virtual Rokhlin property

Extension of classification to mapping class groups of 2-manifolds

Establishment of the virtual Rokhlin property as a significant structural feature

Abstract

We introduce and motivate the definition of the virtual Rokhlin property for topological groups. We then classify the 2-manifolds whose homeomorphism groups have the virtual Rokhlin property. We also establish the analogous result for mapping class groups of 2-manifolds.