Keynote Speaker
Eric Swenson
Brigham Young University
From continua to trees
Abstract: Let Z be a continuum, a compact connected metric space. Z is called m-thick if Z can be separated by removing m points, but cannot be separated by removing less than m points. A subset A ⊆ Z of an m-thick continuum Z is called a min-cut if Z-A is not connected and |A|=m. We will see that for any m-thick continuum there is an ℝ tree T which encodes the separation "structure" of the min-cuts of Z. What this is used for: If you have a group G acting on Z by homeomorphisms, you obtain an action of G on T by homeomorphisms. Under certain conditions on the action of G on Z you obtain a non-nesting action of G on T. This allows us to use the Rips machine to understand the structure of G. | Keynote Speaker
Tim Riley
Cornell University
Soficity and variations on Higman's group
Abstract: A group is sofic when every finite subset can be well approximated in a finite symmetric group. The outstanding question (due to Gromov) is whether every group is sofic. I will give an introduction to this subject and will discuss some recent developments. Helfgott and Juschenko recently argued that a celebrated group constructed by Higman is unlikely to be sofic because its soficity would imply the existence of some seemingly pathological functions. I will describe joint work with Martin Kassabov and Vivian Kuperberg in which we construct variations on Higman's group which have large sofic quotients (in an appropriate sense). By applying Helfgott and Juschenko's arguments, we deduce the existence of similarly pathological functions. This casts doubt on whether Helfgott and Juschenko's heuristics represent evidence for the non-soficity of Higman's group. | Keynote Speaker
Zoran Šunić
Hofstra University
Ordering free groups, trees, and free products, not necessarily in that order
Abstract: We introduce left orders on the free group of any finite rank that are particularly easy to state and work with. The original approach uses actions of the free group on the real line, which is itself an ordered object, and then letting the group just "borrow" the order from the line. We then recover the same orders, and a lot more, through a more organized and sophisticated approach. Namely, we order free products by first ordering their associated Bass-Serre trees and then letting the groups "borrow" the order from the trees. |