Pure Maths Colloquium: Jing Tao

Let \(S\) be a closed orientable surface of genus \(g\). The mapping class group \(\operatorname{MCG}(S)\) of \(S\) is the group of isotopy classes of homeomorphisms of \(S\). In the 1970s, Thurston revolutionized the way we think about mapping class groups. Generalizing the notion of Anosov maps of the torus, he defined pseudo-Anosov maps of higher genus surfaces. He then showed every element of a mapping class group is one of three types: finite order, reducible, or pseudo-Anosov. This is reminiscent of the classification of elements of \(\operatorname{SL}(2,\mathbb{Z})\) into finite order, reducible, or irreducible. While there are these three types, it is natural to wonder which type is more prevalent. In any reasonable way to sample matrices in \(\operatorname{SL}(2,\mathbb{Z})\), irreducible matrices should be generic. One expects something similar for pseudo-Anosov elements in \(\operatorname{MCG}(S)\). 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 \(\operatorname{MCG}(S)\), and show that the proportion of pseudo-Anosov elements in a ball of radius \(r\) tends to 1 as \(r\) tends to infinity. The norms we consider have the commonality that they reflect that \(\operatorname{MCG}(S)\) come from homeomorphisms of S, and can be thought of as the natural analogues of matrix norms on \(\operatorname{SL}(2,\mathbb{Z})\).