Fibring of manifolds and groups

ended

The study of manifolds that fibre over the circle has a long and exciting history at the core of modern manifold topology, starting with Farrell's work on the problem in high ('surgery') dimensions, and running through the celebrated work of Stallings and Thurston in dimension 3, to Agol's 2013 solution of Thurston's virtual fibring conjecture. Parallel developments in group theory have placed the study of Bieri-Neumann-Strebel (BNS) invariants, which emerged in the 1980s, at the heart of the subject; these invariants describe when a group fibres, i.e. admits a map onto Z with finitely generated kernel. In the research outlined here a powerful new set of algebraic invariants - agrarian polytopes - will be used to establish a new frontier in the theory of fibring. The main goal is to achieve a complete description of all possible fibrings over the circle for aspherical manifolds in surgery dimensions. An agrarian polytope is a subset of the vector space H_1(X;R), where X is a group or a manifold. It is defined in the novel framework of agrarian invariants that I am developing, a theory that has already borne remarkable fruit. The theory provides algebraic counterparts to the (analytic) L2-invariants that have proved so powerful in geometric topology, group theory and global analysis over the last four decades. The primary focus of the research proposed here lies in establishing new deep connections between the algebra of group rings and their completions, and global properties of aspherical manifolds and groups. Three further goals of the proposal are: to develop the theory of agrarian invariants in positive characteristic; to show that agrarian invariants are profinitely rigid; to apply the new technology to the study of dynamical zeta functions. Each of these goals promises a breakthrough in its respective domain.