2025/11/20 by Tüli̇n Altunöz, Altunoz, Tulin, Mehmetci̇k Pamuk +3 · 1 citation
Mathematics · Physics and Astronomy · #Analytic and geometric function theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2511.16840
openalex publication_date 2025/11/20 · openalex created_date 2025/11/25 · openalex updated_date 2026/07/28
The mapping class group of an orientable surface, which records its symmetries up to isotopy, plays a central role in low-dimensional topology. This chapter explores the foundational problem of determining minimal generating sets for these groups. We chart the development of this area from classical results involving Dehn twist generators to more recent breakthroughs showing that mapping class groups can be generated by just two elements, pairs of torsion elements, or a small collection of involutions. This chapter contains a discussion of the most current results for punctured surfaces, including a new improvement showing that for an even number of punctures p≥ 8 the group Mod(Σ13,p) is generated by three involutions. Throughout, we highlight the rich interplay between the algebraic features of these generating sets and the underlying geometric structures they encode. The chapter aims to provide a comprehensive account of the pursuit of algebraic and geometric efficiency within one of topology's most intricate and influential groups.