Coarse cohomology, quasimorphisms and Poincaré duality in dimension 3

Margolis recently introduced a framework for coarse homological invariants, generalising group theoretic notions in a way that captures the large-scale geometry of a space. Applied to a PD^3 group G admitting certain quasimorphisms, it can be used to show that G must either arise as the fundamental group of a torus bundle over S^1, or it is finitely presented with a coarse fibration structure that can be used to build a manifold quasiisometric to G and, in the hyperbolic case, induce a faithful circle action. In this talk we will give an introduction to Margolis's coarse cohomology and explain how to use it to obtain the structural properties described above. This talk is based on joint work with Will Thomas.