A group is an algebraic structure capturing the abstract properties of the collection of symmetries of a geometric object or of a physical or mechanical system. The project will concentrate on the study of free-by-cyclic groups. These groups are natural counterparts of the fundamental groups of fibered 3-manifolds, which are important mathematical objects playing key roles in Topology, Differential Geometry, Ergodic Theory, Number Theory, Computational Complexity, and other branches of Mathematics. Compared with the 3-manifold groups, the free-by-cyclic groups are much less well understood, even though they exhibit a considerably wider variety of interesting features and behavior. The project aims to study the geometry and dynamics of free-by-cyclic groups, particularly through exploring new polynomial invariants of these groups. This research should help better understand both the differences and the similarities between the free-by-cyclic groups and their 3-manifold "cousins", and lead to new insights about the interactions of algebra and dynamics in group theory.
The project will build on the new work of the proposer, joint with Leininger and Dowdall, studying the properties of free-by-cyclic groups by means of a "folded mapping torus", associated to such a group. The folded mapping torus comes equipped with a natural semi-flow of continuous transformations, and it encodes in a particularly useful and accessible form key algebraic, geometric and dynamical information about the group. Another key object, associated to the folded mapping torus, is the "McMullen polynomial", which is a free-by-cyclic analog of the Teichmuller polynomial for fibered 3-manifolds. The main overall research goal of the project is to understand algebraic, geometric and dynamical features related to the many possible ways in which a free-by-cyclic group G can split as a free-by-cyclic group and as an ascending HNN-extension of a free group, and to gain new knowledge about free group automorphisms. Specific problems to be studied include: understanding the relationship between the "cone of sections" of folded mapping torus with the BNS-invariant of the group, uniformizing the stable trees and the Cannon-Thurston maps associated with different splittings of G, studying irreducibility properties of monodromy automorphisms associated to different free-by-cyclic splittings of G, developing an analog of the homological zeta-function for G, and others.
