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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.
Keywords: geometric group theory, quasi-isometry, group cohomology, cohomological dimension, Poincaré duality group, continuous orbit equivalence
Li, Xin 1
@article{10_2140_agt_2018_18_3477,
author = {Li, Xin},
title = {Dynamic characterizations of quasi-isometry and applications to cohomology},
journal = {Algebraic and Geometric Topology},
pages = {3477--3535},
publisher = {mathdoc},
volume = {18},
number = {6},
year = {2018},
doi = {10.2140/agt.2018.18.3477},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.3477/}
}
TY - JOUR AU - Li, Xin TI - Dynamic characterizations of quasi-isometry and applications to cohomology JO - Algebraic and Geometric Topology PY - 2018 SP - 3477 EP - 3535 VL - 18 IS - 6 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.3477/ DO - 10.2140/agt.2018.18.3477 ID - 10_2140_agt_2018_18_3477 ER -
%0 Journal Article %A Li, Xin %T Dynamic characterizations of quasi-isometry and applications to cohomology %J Algebraic and Geometric Topology %D 2018 %P 3477-3535 %V 18 %N 6 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.3477/ %R 10.2140/agt.2018.18.3477 %F 10_2140_agt_2018_18_3477
Li, Xin. Dynamic characterizations of quasi-isometry and applications to cohomology. Algebraic and Geometric Topology, Tome 18 (2018) no. 6, pp. 3477-3535. doi: 10.2140/agt.2018.18.3477
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