Dynamic characterizations of quasi-isometry and applications to cohomology

We build a bridge between geometric group theory and topological dynamical systems by establishing a dictionary between coarse equivalence and continuous orbit equivalence. As an application, we show that group homology and cohomology in a class of coefficients, including all induced and coinduced modules, are coarse invariants. We deduce that being of type [math] (over arbitrary rings) is a coarse invariant, and that being a (Poincaré) duality group over a ring is a coarse invariant among all groups which have finite cohomological dimension over that ring. Our results also imply that every coarse self-embedding of a Poincaré duality group must be a coarse equivalence. These results were only known under suitable finiteness assumptions, and our work shows that they hold in full generality.