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
(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.
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