Dynamic characterizations of quasi-isometry, and applications to cohomology

Abstract

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 $F{P}_n$ (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 self coarse embedding of a Poincaré duality group over an arbitrary ring must be a coarse equivalence.

29 pages; improved results and exposition; added and updated references

References in corpus (8)

• Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras
• Uncountably many groups of type $FP$
• Cantor systems and quasi-isometry of groups
• Continuous orbit equivalence rigidity
• Vanishing of ${\ell }^2$-Betti numbers of locally compact groups as an invariant of coarse equivalence
• Partial transformation groupoids attached to graphs and semigroups
• From the geometry of box spaces to the geometry and measured couplings of groups
• Sofic boundaries of groups and coarse geometry of sofic approximations