Abstracts

Sahana Balasubramanya - Hyperbolic structures on groups
Abstract: For every group G, we introduce the set of hyperbolic structures on G, denoted H(G), which consists of equivalence classes of (possibly infinite) generating sets of G such that the corresponding Cayley graph is hyperbolic; two generating sets of G are equivalent if the corresponding word metrics on G are bi-Lipschitz equivalent. Alternatively, one can define hyperbolic structures in terms of cobounded G-actions on hyperbolic spaces. We are especially interested in the subset AH(G) ⊆ H(G) of acylindrically hyperbolic structures on G, i.e., hyperbolic structures corresponding to acylindrical actions. Ele- ments of H(G) can be ordered in a natural way according to the amount of information they provide about the group G. The main goal of our work is to initiate the study of the posets H(G) and AH(G) for various groups G. We discuss basic properties of these posets such as cardinality and existence of extremal elements, obtain several re- sults about hyperbolic structures induced from hyperbolically embedded subgroups of G, and study the question to what extent a hyperbolic structure is determined by the set of loxodromic elements and their translation lengths.

Juliette Bavard - Quasimorphisms on a big mapping class group
Abstract: The mapping class group of the plane minus a Cantor set naturally appears in many dynamical contexts, including group actions on surfaces, the study of groups of homeomorphisms on a Cantor set, and complex dynamics. In this talk, I will explain how to construct quasimorphisms on this big mapping class group.

Clark Butler - Characterizing locally symmetric spaces by their Lyapunov exponents
Abstract: We show that closed negatively curved locally symmetric spaces are characterized among nearby Riemannian manifolds by the Lyapunov exponents of their geodesic flow with respect to volume. Our methods extend to locally characterize these geodesic flows by their volume exponents even among arbitrary smooth perturbations of the dynamical system. We also discuss some notions of partial rigidity for the geodesic flows of the variable negative curvature locally symmetric spaces. Finally we introduce some new dimensional invariants for these flows which are closely related to the 1/4-pinching rigidity theorem for complex hyperbolic manifolds.

Sam Corson - Some new results in automatic continuity
Abstract: Diagonalization arguments have been used to show that homomorphisms from groups admitting infinite multiplication to groups similar to free groups are always boring. We give a generalized criterion that extends many results in this vein. Groups including torsion-free word hyperbolic groups and Thompson's group F satisfy this criterion.

Matthew Haulmark - A classification theorem for 1-dimensional boundaries of groups with isolated flats
Abstract: In 2000 Kapovich and Kleiner proved that if G is a one-ended hyperbolic group that does not split over a two-ended subgroup, then the boundary of G is either a Menger curve, a Sierpinski carpet, or a circle. Kim Ruane observed that there were no known non-hyperbolic examples of groups with Menger curve boundary, and asked if there was a CAT(0) generalization of Kapovich and Kleiner's theorem. As boundaries of CAT(0) groups are in general not locally connected, there is no hope of such a generalization for general CAT(0) groups. However, a version of Kapovich and Kleiner's theorem may hold for certain classes of CAT(0) groups. In this talk I will discuss a generalization of the Kapovich-Kleiner theorem for CAT(0) groups with isolated flats, and provide an example of a non-hyperbolic CAT(0) group with Menger curve boundary.

Kasia Jankiewicz - Graph coloring problem and fibering right angled Coxeter groups
Abstract: I will describe a simple graph coloring problem whose solution, if exists, provides a virtual algebraic fibration of the right angled Coxeter group associated to the graph. I will discuss some examples of graphs with or without solutions. I will give an introduction to Bestvina-Brady Morse theory on which our results rely. This is joint work with Sergey Norin and Daniel Wise.

Franco Vargas Palette - Local and global minimum of Renormalized Volume
Abstract: The renormalized volume $V_R$ is a finite quantity associated to geometrically finite hyperbolic 3-manifolds of infinite volume. In this talk I'll discuss its definition and some properties for convex co-compact manifolds, such as local convexity and convergence under suitable limits.

Andrew Putman - Finite generation of the Johnson filtration
Abstract: I will prove that the kth term of the Johnson filtration of the mapping class group is finitely generated for g>>k. This is joint work with Tom Church and Mikhail Ershov.

Andrew Zimmer - Gromov hyperbolicity in several complex variables
Abstract: In this talk I will describe how Gromov hyperbolic metric spaces appear naturally in several complex variables and then give some applications. The talk will assume no background in several complex variables.