Geometry, group theory, and dynamics

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

Abstract Award: DMS 1510034, Principal Investigator: Christopher J. Leininger To study the long-term behavior of an evolving physical or geometric system one can look at its "cross sections." For example, to analyze a fluid flowing throughout a closed system, we might deposit tracers into the fluid at a particular location then examine how it has been mixed, distorted, or changed when (and if) it returns. We can also try to predict how the behavior of the "first-return" to the starting location-called a cross section-might influence or predict the behavior at some other cross section. One goal of this project is to study the evolution of certain classes of abstract mathematical systems, similar in many ways to the fluid flow just described, which arise from geometric and algebraic considerations. Previous work of the PI with S. Dowdall and I. Kapovich provides structural results which relate the cross sections and their first-returns to one another and to the original system. The PI will continue this analysis with Dowdall and Kapovich and pursue deeper connections between different cross sections and with the ambient system. The PI's work with Dowdall and Kapovich, past and future, is motivated by the foundational work of W. Thurston, D. Fried, and C. McMullen in a more classical setting, but diverges in fundamental and striking ways. The PI will also continue his analysis of the classical setting with Agol and Margalit investigating further connections and generalizations that might provide a unified theory. This project involves aspects of geometry, topology, group theory, and dynamics. The central objects are surface homeomorphisms and train track maps for free groups automorphisms. The PI will continue his work with Dowdall and Kapovich, analyzing free-by-cyclic groups via special semi-flows on 2-complexes. In addition to strengthening the connection between different cross sections, for example proving that full-irreducibility for monodromies is a property shared by all or none of the sections, he will also tie together all the monodromies with an action of the entire group on a tree and connect the Cannon-Thurston maps to each other. With Margalit and Agol-Margalit, the PI will continue to analyze the classical setting of suspension flows of pseudo-Anosov homeomorphisms in an attempt to understand all systoles of moduli spaces. In his work with Kent, Bestvina-Bromberg-Kent, and others, the PI will continue his geometric analysis of subgroups of the mapping class group and convex cocompactness.