# Pure Maths Colloquium: Jing Tao

This talk is part of the Pure Maths Colloquium at the University of St Andrews. Check out our upcoming talks at https://theran.lt/pure-colloquium/.

 Where: Zoom 894 5212 7915 When: Oct 28 2021 @ 16.00 Video: Link (internal)
 Speaker: Jing Tao University of Oklahoma Title: Genericity of pseudo-Anosov mapping classes

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 $$\operatorname{SL}(2,\mathbb{Z})$$.