Type: Article

Conjugacy classes of big mapping class groups

Hernández Hernández, Jesús; Hrušák, Michael; Morales, Israel; Randecker, Anja; Sedano, Manuel; Valdez, Ferrán;

Abstract:

This paper considers big mapping class groups of infinite-type surfaces. The main results provide a characterization of conjugacy classes with respect to dense/meager properties. The authors describe a detailed correspondence between results in automorphism groups of countable structures and conjugacy actions of big mapping class groups via Fraïssé theory and curve graphs. The paper paints a complete story of how the model-theoretical results by A. S. Kechris and C. Rosendal [Proc. Lond. Math. Soc. (3) 94 (2007), no. 2, 302–350; MR2308230] and J. K. Truss [Proc. London Math. Soc. (3) 65 (1992), no. 1, 121–141; MR1162490] can first be translated to geometric embedding properties, and then used to study conjugacy classes of subgroups of the big mapping class group. Dehn twists, non-displaceable subsurfaces, and maximal ends are the main geometric viewpoints that allow the main results to be illustrated through key examples and the proofs to be presented with clear steps. The authors do a phenomenal job in the presentation of their results. The background section provides carefully crafted introductions to the topics of infinite-type surfaces, big mapping class groups, curve graphs, Fraïssé theory, and Fraïsséfication. However, prior knowledge in mapping class groups would be greatly helpful. The proofs of the main results are neatly broken down into bite-size arguments, which also allows the authors to place narratives, comments, and remarks to smoothen the flow of the paper. A list of open questions is provided for any intrigued readers who want more to think about. Overall, the paper is of superb quality, and is definitely worth the time of anyone in the field. The examples would be perfect for anyone learning the field