Georgia Institute of Technology College of Sciences

I read Benoist's paper Convexes Divisibles IV (2006, Invent. Math.), and will talk about it. The main result is a striking structural theorem for triangles in the boundaries of 3-dimensional properly convex divisible domains O that are not strictly convex (which exist). These bound "flats" in O. Benoist shows that every Z^2 subgroup of the group G preserving O preserves a unique such triangle. Conversely, all such triangles are disjoint and any such triangle descends to either a torus or Klein bottle in the quotient M = O/G (and so must have many symmetries!). Furthermore, this "geometrizes" the JSJ decomposition of M, in the sense that cutting along these tori and Klein bottles gives an atoroidal decomposition of M.

Virtual knot theory is a variant of classical knot theory in which one allows a new type of crossing called a "virtual" crossing. It was originally developed by Louis Kauffman in order to study the Jones polynomial but has since developed into its own field and has genuine significance in low dimensional topology. One notable interpretation is that virtual knots are equivalent to knots in thickened surfaces. In this talk we'll introduce virtual knots and show why they are a natural extension of classical knots. We will then discuss what virtual knot theory can tell us about the both the classical Jones polynomial and its potential extensions to knots in arbitrary 3-manifolds. An important tool we will use throughout the talk is the knot quandle, a classical knot invariant which is complete up to taking mirror images.

The curve graph provides a combinatorial perspective to study surfaces. Classic work of Ivanov showed that the automorphisms of this graph are naturally isomorphic to the mapping class group. By dropping isotopies, more recent work of Long-Margalit-Pham-Verberne-Yao shows that there is also a natural isomorphism between the automorphisms of the fine curve graph and the homeomorphism group of the surface. Restricting this graph to smooth curves might appear to be the appropriate object for the diffeomorphism group, but it is not. In this talk, we will discuss why this doesn’t work and some progress towards describing the group of homeomorphisms that is naturally isomorphic to automorphisms of smooth fine curve graphs.

The fine curve graph of a surface is a graph whose vertices are essential simple closed curves in the surface and whose edges connect disjoint curves. Following a rich history of hyperbolicity in various graphs based on surfaces, the fine curve was shown to be hyperbolic by Bowden–Hensel–Webb. Given how well-studied the curve graph and the case of “up to isotopy” is, we ask: what about the mysterious part of the fine curve graph not captured by isotopy classes? In this talk, we introduce the result that the subgraph of the fine curve graph spanned by curves in a single isotopy class is not hyperbolic; indeed, it contains a flat of EVERY dimension. Along the way, we will discuss how to not prove this theorem as we explore proofs of hyperbolicity of related complexes. This work is joint with Ryan Dickmann.