|DATE||July 29 (Wed), 2020|
|TITLE||Mapping class groups in complex dynamics|
In joint work with James Belk, Justin Lanier and Becca Winarski, we give a simple geometric algorithm that can be used to determine whether or not a post-critically finite topological polynomial is Thurston equivalent to a polynomial. Our methods are rooted in geometric group theory: we consider a complex of isotopy classes of trees and a simplicial map of this complex to itself that we call the lifting map. Our work extends previous work of Nekrashevych. Similar work has been announced by Ishii-Smillie. We will give several applications of our methods, including a solution to Pilgrim's finite global attractor problem in the case of topological polynomials, the solution to a generalization of Hubbard’s twisted rabbit problem (originally solved by Bartholdi–Nekrashevych), and a new proof of Thurston's theorem for topological polynomials.