  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 give conceptual explanations for
previous results of Shalom and Sauer on coarse invariance of homological and
cohomological dimensions and Shalom's property $H_{FD}$. As another
application, we show that group homology and cohomology in a class of
coefficients, including all induced and co-induced modules, are coarse
invariants. We deduce that being of type $FP_n$ (over arbitrary rings) is a
coarse invariant, and that being a (Poincar\'e) 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 self coarse embedding of a
Poincar\'e duality group over an arbitrary ring must be a coarse equivalence.
