Algebraic Structure of Mapping Class Groups via Dynamics

Random walk on the mapping class group produces mapping classes with large translation lengths on Teichmüller space. In 2007, Maher and Tiozzo showed that they are not normal generators of the mapping class group. In contrast, Lanier and Margalit proved in 2018 that pseudo-Anosov mapping classes with small translation lengths are normal generators of the mapping class group. In this talk, we introduce related work and show that reducible mapping classes with small translation lengths also normally generate the mapping class group using simple combinatorics. This is joint work with Hyungryul Baik and Chenxi Wu. We also discuss an analogous question for asymptotic translation length on curve graphs and related work if time permits.