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2018 Dynamic characterizations of quasi-isometry and applications to cohomology
Xin Li
Algebr. Geom. Topol. 18(6): 3477-3535 (2018). 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 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.

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Xin Li. "Dynamic characterizations of quasi-isometry and applications to cohomology." Algebr. Geom. Topol. 18 (6) 3477 - 3535, 2018. https://doi.org/10.2140/agt.2018.18.3477

Information

Received: 30 October 2017; Revised: 10 May 2018; Accepted: 7 June 2018; Published: 2018
First available in Project Euclid: 27 October 2018

zbMATH: 06990070
MathSciNet: MR3868227
Digital Object Identifier: 10.2140/agt.2018.18.3477

Subjects:
Primary: 20F65, 20J06
Secondary: 37B99

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.18 • No. 6 • 2018
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