Hyperbolicity for Schmutz graph of degree n
For a given surface, let G be a simplicial complex whose vertices are isotopy classes of nonseparating simple closed curves and two vertices are adjacent iff their intersection number is n. By showing G is quasi-isometric to the curve complex, we will show G is uniformly hyperbolic for each n. In a special case, we can get a smaller hyperbolicity constant for a Schmutz graph using unicorn paths in arc complex.
Funda Gultepe (UIUC)
Fully irreducible Automorphisms via Dehn twisting in #_n(S^2 X S^1)
By using geometric interpretation of a Dehn twist in #_n(S^2 X S^1), we show that when two free factors are far enough from each other in the free factor complex, the corresponding Dehn twist automorphisms generate a free group of rank 2. Moreover, every element from this group is either conjugate to a power of one of the Dehn twists or it is a fully irreducible automorphism.
Priyam Patel (Purdue)
Effective Separability for Hyperbolic Surface and 3-Manifold Groups
The fundamental groups of hyperbolic surfaces and 3-manifolds, referred to as surface groups and 3-manifold groups, respectively, have various algebraic finiteness properties. Two of these properties, residual finiteness and subgroup separability, have played an important role in the recent resolution of some outstanding conjectures in 3-manifold theory. In this talk, we will explain how effective proofs of these properties can help us "quantify" separability and discuss the topological implications of such quantifications. We initially focus on the hyperbolic surface case and then discuss joint work with K. Bou-Rabee and M.F. Hagen extending effective separability results to hyperbolic 3-manifold groups using right-angled Artin groups.
Dominic Dotterrer (University of Chicago)
Geometric and Topological Embeddings Simplicial Complexes
One of the geometric applications of expander graphs is that they provide an example of a worst case in Bourgain's embedding algorithm. They are also examples of sparse graphs which only embed into surfaces with very large genus. It seems natural, therefore, to ask how expander properties of higher dimensional simplicial complexes might pose difficulties in embedding problems. I will state some theorems and some open problems.
Anton Lukyanenko (University of Michigan)
Complex hyperbolic space and Heisenberg continued fractions
Complex hyperbolic space CH is a rank one symmetric space defined analogously to the familiar real hyperbolic space RH, but over the complex numbers. It is negatively curved (though with variable curvature), and its boundary is naturally identified with the compactified Heisenberg group, a natural generalization of the real and complex numbers which serve as boundaries for RH^2 and RH^3, respectively. The simplest lattice in the isometry group of CH is known as the Picard modular group, and is an immediate generalization of the classic modular group PSL(2,Z). Just as the study of PSL(2,Z) is closely tied to continued fractions on the real line, studying the Picard modular group leads to a notion of continued fractions on the Heisenberg group. In the talk, I will describe the geometry of these various spaces, and touch on the dynamics of the continued fraction theory.