Dynamic characterizations of quasi-isometry, and applications to cohomology
Algebraic and Geometric Topology
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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 co-induced modules, are coarse invariants. We deduce that being of type FPn (over arbitrary rings) is a coarse invariant, and that being a (Poincare) 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 Poincare 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.
- Pure Mathematics