In the world of finite-type surfaces, one of the key tools to studying the mapping class group is to study its action on the curve graph. The curve graph is a combinatorial object intrinsic to the surface, and its appeal lies in the fact that it is infinite-diameter and $\delta$-hyperbolic. For infinite-type surfaces, the curve graph disappointingly has diameter 2. However, all hope is not lost!
Surfaces of infinite type, such as the plane minus a Cantor set, occur naturally in dynamics. However, their mapping class groups are much less studied and understood compared to the mapping class groups of surfaces of finite type. Many fundamental questions remain open. We will discuss the mapping class group G of the plane minus a Cantor set, and show that any nontrivial G-action on the circle is semi-conjugate to its action on the so-called simple circle.
Fibered knots in a three-manifold Y can be thought of as the binding of an open book decomposition for Y. A basic question to ask is how properties of the open book decomposition relate to properties of the corresponding knot.
All 3-manifolds bound 4-manifolds, and many construction of 3-manifolds automatically come with a 4-manifold bounding it. Often times these 4-manifolds have definite intersection form. Using Heegaard Floer correction terms and an analysis of short characteristic covectors in bimodular lattices, we give an obstruction for a 3-manifold to bound a definite 4-manifold, and produce some concrete examples. This is joint work with Kyle Larson.
The writhe of a braid (=#pos crossing - #neg crossings) and the fractional Dehn twist coefficient of a braid (a rational number that measures "how much the braid twists") are the two most prominent examples of what is known as a quasimorphism (a map that fails to be a group homomorphism by at most a bounded amount) from Artin's braid group on n-strands to the reals.
In 2010, Bestvina-Bromberg-Fujiwara proved that the mapping class group of a finite type surface has finite asymptotic dimension. In contrast, we will show the mapping class group of an infinite-type surface has infinite asymptotic dimension if it contains an essential shift. This work is joint with Curtis Grant and Kasra Rafi.
Given a knot K in the 3-sphere, the 4-genus of K is the minimal genus of an orientable surface embedded in the 4-ball with boundary K. If the knot K has a symmetry (e.g. K is periodic or strongly invertible), one can define the equivariant 4-genus by only minimising the genus over those surfaces in the 4-ball which respect the symmetry of the knot. I'll discuss some work with Keegan Boyle trying to understanding the equivariant 4-genus.
Cabling is one of important knot operations. We study various properties of cable knots and how to characterize the cable knots by its complement.
Bestvina--Brady groups are subgroups of right-angled Artin groups, and their Dehn functions are bounded above by quartic functions. There are examples of Bestvina--Brady groups whose Dehn functions are linear, quadratic, cubic, and quartic. In this talk, I will give a class of Bestvina--Brady groups that have polynomial Dehn functions, and we can identify the Dehn functions by the defining graphs of those Bestvina--Brady groups.
