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Dynamic characterizations of quasi-isometry and applications to cohomology

Xin Li

Algebraic & Geometric Topology 18 (2018) 3477–3535

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 FPn (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.

geometric group theory, quasi-isometry, group cohomology, cohomological dimension, Poincaré duality group, continuous orbit equivalence
Mathematical Subject Classification 2010
Primary: 20F65, 20J06
Secondary: 37B99
Received: 30 October 2017
Revised: 10 May 2018
Accepted: 7 June 2018
Published: 18 October 2018
Xin Li
School of Mathematical Sciences
Queen Mary University of London
United Kingdom