Dynamic characterizations of quasi-isometry, and applications to cohomology

TL;DR

This paper establishes a connection between coarse geometric properties and topological dynamical systems, showing that various cohomological invariants are preserved under coarse equivalences, with applications to group theory.

Contribution

It introduces a new framework linking coarse invariance with topological dynamics and demonstrates that several algebraic and cohomological properties are coarse invariants.

Findings

Homological and cohomological dimensions are coarse invariants.

Type FP_n and Poincaré duality are coarse invariants.

Self coarse embeddings of Poincaré duality groups are coarse equivalences.

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 $FP_n$ (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.