Structure and Topological Generation of Big Mapping Class Groups

TL;DR

This paper explores the structure and generation properties of big mapping class groups associated with infinite-type surfaces, highlighting key differences from finite-type cases and summarizing recent foundational results.

Contribution

It provides a systematic overview of the structure and topological generation of big mapping class groups, emphasizing new insights and differences from classical finite-type surface groups.

Findings

Finite topological generation results for certain infinite-type surfaces

Clear exposition of structural differences from finite-type mapping class groups

Summary of foundational results in the field

Abstract

Big mapping class groups are the mapping class groups of infinite-type surfaces, that is, surfaces whose fundamental groups are not finitely generated. While mapping class groups of finite-type surfaces have been extensively studied, the theory of big mapping class groups is a recent and rapidly developing area of research. This thesis provides a systematic introduction to the structure and topological generation of big mapping class groups, emphasizing the key differences from the classical finite-type case. It presents a clear exposition of foundational results concerning their topological and algebraic structure, including known results on finite topological generation for a certain family of infinite-type surfaces.