Copenhagen
Groups and Operator Algebras Seminar

Abstract:  Let S be a surface of finite type. By Nielsen-Thurston Classification, every element of the mapping class group Map(S) of S is either finite order, reducible, or pseudo-Anosov. While there are these three types, it seems that from any reasonable point of view a “generic” element of Map(S) is pseudo-Anosov. In joint work with Erlandsson and Souto, we define a notion of genericity and show that pseudo-Anosov elements are indeed generic. More precisely, we consider several “norms” on Map(S), and show that the proportion of pseudo-Anosov elements in a ball of radius r tends to 1 as r tends to infinity. These norms have the commonality that they reflect that Map(S) come from homeomorphisms of S, and they can be thought of as natural analogues of matrix norms.