The cohomology of a group G with coefficients in the group
ring ZG has a natural ZG-module structure. It turns out that
under appropriate finiteness assumptions, the set of subgroups
in G that can occur as the stabilizer of some Z-submodule
is, in a natural sense, quasi-isometry invariant. After
explaining some applications of this result to quasi-isometric
rigidity, I will explain how special structure in the second
cohomology leads to:
-A proof of a conjecture of Dunwoody-Dicks characterizing
virtual surface groups.
-A cohomological characterization of the JSJ decomposition
of Dunwoody-Sageev/Dunwoody-Swenson.
-A new proof of the theorem of Papasoglu on the quasi-isometry
invariance of the JSJ decomposition.