Dynamic characterizations of quasi-isometry and applications to cohomology

Li, Xin

doi:10.2140/agt.2018.18.3477

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Dynamic characterizations of quasi-isometry and applications to cohomology

Xin Li

Algebraic & Geometric Topology 18 (2018) 3477–3535

DOI: 10.2140/agt.2018.18.3477

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 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 ${FP}_{n}$ (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 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.

Keywords

geometric group theory, quasi-isometry, group cohomology, cohomological dimension, Poincaré duality group, continuous orbit equivalence

Mathematical Subject Classification 2010

Primary: 20F65, 20J06

Secondary: 37B99

References

Publication

Received: 30 October 2017

Revised: 10 May 2018

Accepted: 7 June 2018

Published: 18 October 2018

Authors

Xin Li
School of Mathematical Sciences Queen Mary University of London London United Kingdom

Volume 18, issue 6 (2018)